diff --git "a/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/2301.03055v1.pdf.txt" "b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/2301.03055v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/2301.03055v1.pdf.txt" @@ -0,0 +1,4884 @@ +Spectral estimates for free boundary minimal +surfaces via Montiel–Ros partitioning methods +Alessandro Carlotto, Mario B. Schulz, David Wiygul +Abstract +We adapt and extend the Montiel–Ros methodology to compact manifolds with boundary, +allowing for mixed (including oblique) boundary conditions and also accounting for the action of +a finite group G together with an additional twisting homomorphism σ: G → O(1). We then +apply this machinery in order to obtain quantitative lower and upper bounds on the growth rate +of the Morse index of free boundary minimal surfaces with respect to the topological data (i. e. +the genus and the number of boundary components) of the surfaces in question. In particular, we +compute the exact values of the equivariant Morse index and nullity for two infinite families of +examples, with respect to their maximal symmetry groups, and thereby derive explicit two-sided +linear bounds when the equivariance constraint is lifted. +1 Introduction +Despite a profusion of constructions of free boundary minimal surfaces in the Euclidean unit ball +B3 over the course of the past decade ([14–16,24,31] via optimization of the first Steklov eigenvalue, +[4,25,26] via min-max methods for the area functional, and [6,11,18–20,22] via gluing methods), +many basic questions about the space of such surfaces remain open. The reader is referred to +[12, 13, 27] for recent overviews of the field. In particular, so far it is only for the rotationally +symmetric examples, planar discs through the origin and critical catenoids, that the exact value of +the Morse index is actually known (see [8,36,38]). The present manuscript is the first in a series of +works aimed at shedding new light on this fundamental invariant, which (also due to its variational +content, and thus to its natural connection with min-max theory, cf. [28–30] and references therein) +has acquired great importance within geometric analysis. +Partly motivated by the corresponding conjectures concerning closed minimal hypersurfaces in +manifolds of positive Ricci curvature (cf. [1, 33]), five years ago the first-named author proved +with Ambrozio and Sharp a universal lower bound for the index of any free boundary minimal +surface in any mean-convex subdomain Ω of R3 in terms of the topological data of the surface under +consideration. Specifically, it was shown in [2] that the following estimate holds: +index(Σ) ≥ 1 +3(2g + b − 1) +(1.1) +where Σ is any free boundary minimal surface in Ω, and g, b denote respectively its genus and the +number of its boundary components. It is then a natural, and by now well-known question, whether +such a lower bound can be complemented by an affine upper bound or whether – instead – it is +1 +arXiv:2301.03055v1 [math.DG] 8 Jan 2023 + +1 Introduction +A. Carlotto, M. B. Schulz, D. Wiygul +conceivable to have a superlinear growth rate of the index with respect to g and b. In this article we +show that there are in fact infinite families of free boundary minimal surfaces in B3 whose index is +bounded from above (and below) by explicit affine functions of the topological data. More broadly, +we embed such a result in a network of index estimates that in turn build on a generalization of the +fundamental Montiel–Ros methodology – as first presented in [32] – that is of independent interest +and wider applicability. +In general terms, we shall be concerned here with proving effective estimates for (part of) the +spectrum of Schrödinger-type operators on bounded Lipschitz domains of Riemannian manifolds, +combined with mixed boundary conditions, that will be – on disjoint portions of the boundary in +question – of Dirichlet or Robin (oblique) type. Summarizing and oversimplifying things to the +extreme, the number of eigenvalues of any such operator below a given threshold can be estimated +by suitably partitioning the domain into finitely many subdomains, provided one adjoins Dirichlet +boundary conditions in the interior boundaries when aiming for lower bounds, and Neumann +boundary conditions in the interior boundaries for upper bounds instead. We refer the reader to +Section 2 for the setup of our problem together with our standing assumptions, and to the first part +of Section 3 (specifically to Theorem 3.1, and Corollary 3.2) for precise statements. +In fact, often times (yet not always) the partitions mentioned above naturally relate to the underlying +symmetries of the problem in question, which is in particular the case for some of the classes of +free boundary minimal surfaces in B3 that have so far been constructed. With this remark in +mind, a peculiar (and, a posteriori, fundamental) feature of our work is the development of the +Montiel–Ros methodology in the presence of the action of a group G together with an additional +twisting homomorphism σ: G → O(1), in the terms explained in Section 2.4. This allows, for +instance, to explicitly and transparently study how the Morse index of a given free boundary minimal +surface depends on the symmetries one imposes, namely to look at the “functor” (G, σ) → indσ +G(T), +where T denotes the index (Jacobi) form of the surface in question. As apparent even from the +simplest examples we shall discuss, this perspective turns out to be very natural and effective in +tackling the geometric problems we are interested in. +With this approach, lower bounds are sometimes relatively cheap to obtain. One way they can +derived is from ambient Killing vector fields, once it is shown that the associated (scalar-valued) +Jacobi field on the surface under consideration vanishes along the (interior) boundary of any domain +of the chosen partition, which in practice amounts to suitably designing the partition and picking the +Killing field given the geometry of the problem. We present one simple yet paradigmatic such result +in Proposition 4.2, which concerns free boundary minimal surfaces with pyramidal or prismatic +symmetry in B3. Instead, upper bounds are often a lot harder to obtain and shall typically rely on +finer information than the sole symmetries of the scene one deals with. Said otherwise, one needs to +know how (i. e. by which method) the surface under study has been obtained. +We will develop here a detailed analysis of the Morse index of the two families of free boundary +minimal surfaces we constructed in our recent, previous work [6]. Very briefly, using gluing methods +of essentially PDE–theoretic character, we obtained there a sequence Σ−K0∪B2∪K0 +m +of surfaces having +genus m, three boundary components and antiprismatic symmetry group Am+1, and a sequence +Ξ−K0∪K0 +n +of surfaces having genus zero, n + 2 boundary components and prismatic symmetry group +Pn. As we described at length in Section 7 therein, with data (cf. Table 2 and Table 3) and +heuristics, numerical simulations for the Morse index of the surfaces in the former sequence display +a seemingly “erratic” behaviour, as such values do not align on the graph of any affine function, nor +seem to exhibit any obvious periodic pattern. This is a rather unexpected behaviour (by comparison +2 + +1 Introduction +A. Carlotto, M. B. Schulz, D. Wiygul +e. g. with other families of examples, say in the round three-dimensional sphere, see [21]), which +obviously calls for a careful study that we carry through in Section 5 of the present article. In +particular, we establish the following statement: +Theorem 1.1 (Index estimates for Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +). There exist m0, n0 > 0 such that +for all integers m > m0 and n > n0 the Morse index and nullity of the free boundary minimal +surfaces Σ−K0∪B2∪K0 +m +, Ξ−K0∪K0 +n +⊂ B3 satisfy the bounds +2m + 1 ≤ ind(Σ−K0∪B2∪K0 +m +), +ind(Σ−K0∪B2∪K0 +m +) + nul(Σ−K0∪B2∪K0 +m +) ≤ 10m + 10, +2n + 2 ≤ ind(Ξ−K0∪K0 +n +), +ind(Ξ−K0∪K0 +n +) + nul(Ξ−K0∪K0 +n +) ≤ 8n. +To the best of our knowledge, this is the very first upper bound obtained for the Morse index of a +sequence of free boundary minimal surfaces in the Euclidean unit ball B3. In fact, the upper bound +in this “absolute estimate” follows quite easily by combining the “relative estimate”, associated to +the equivariant Morse index of these surfaces (with respect to their respective maximal symmetry +groups) with the aforementioned Proposition 3.1. +The next statement thus pertains to such +equivariant bounds, for which we do obtain equality, thus settling part of Conjecture 7.7 (iv) and +Conjecture 7.9 (iv) of [6]. We stress that neither family is constructed variationally, and thus +there is actually no cheap index bound one can extract from the design methodology itself; on +the contrary, this statement indicates a posteriori that the families of surfaces in question may in +principle be constructed (even in a non-asymptotic regime) by means of min-max schemes generated +by 2-parameter sweepouts, modulo the well-known problem of fully controlling the topology in the +process (cf. [4]). +Theorem 1.2 (Equivariant index and nullity of Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +). There exist m0, n0 > 0 +such that for all integers m > m0 and n > n0 the equivariant Morse index and nullity of the free +boundary minimal surfaces Σ−K0∪B2∪K0 +m +, Ξ−K0∪K0 +n +⊂ B3 satisfy +indAm+1(Σ−K0∪B2∪K0 +m +) = 2, +nulAm+1(Σ−K0∪B2∪K0 +m +) = 0, +indPn(Ξ−K0∪K0 +n +) = 2, +nulPn(Ξ−K0∪K0 +n +) = 0. +The main idea behind the proof of these results, or – more precisely – for the upper bounds can +only be explained by recalling, in a few words, how the surfaces in question have been constructed. +Following the general methodology of [17], one first considers a singular configuration, that is a +formal union of minimal surfaces in B3 (not necessarily free boundary), then its regularization +– which needs the use of (wrapped) periodic minimal surfaces in R3, to desingularize near the +divisors, and controlled interpolation processes between the building blocks in play – and, thirdly +and finally, the perturbation of such con��gurations to exact minimality (at least for some values of +the parameters), while also ensuring proper embeddedness and accommodating the free boundary +condition. Here we first get a complete understanding of the index and nullities of the building +blocks, for the concrete cases under consideration in Section 5. In somewhat more detail, the +analysis of the Karcher–Scherk towers (the periodic building blocks employed in either construction) +exploits, in a substantial fashion, the use of the Gauss map, which allows one to rephrase the initial +geometric question into as one for the spectrum of simple elliptic operators of the form ∆gS2 + 2 +on suitable (typically singular, i. e. spherical triangles, wedges or lunes) subdomains of round S2, +3 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +with mixed boundary conditions, and possibly subject to additional symmetry requirements. The +analysis of the other building blocks – disks and asymmetric catenoidal annuli – is more direct, +although, in the latter case, trickier than it may first look (see e. g. Lemma 5.8). +Once that preliminary analysis is done, we then prove that, corresponding to the (local) geometric +convergence results (that are implied by the very gluing methodology) there are robust spectral +convergence results that serve our scopes. However, a general challenge in the process is that +gluing constructions typically have transition regions where different scales interact with each +another: in our constructions of the sequences Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +such regions occur between +the catenoidal annuli K0 (as well as the disk B2 in the former case) and the wrapped Karcher– +Scherk towers, roughly at distances between m−1 and m−1/2 (respectively n−1 and n−1/2) from the +equatorial S1. As a result, we need to deal with delicate scale-picking arguments, an ad hoc study of +the geometry of such regions (cf. Lemma 5.21) and – most importantly – prove the corresponding +uniform bounds for eigenvalues and eigenfunctions (collected in Lemma 5.25), which allow to rule +out pathologic concentration phenomena, thereby leading to the desired conclusions. +Acknowledgements. +The authors wish to express their sincere gratitude to Giada Franz for a +number of conversations on themes related to those object of the present manuscript. This project +has received funding from the European Research Council (ERC) under the European Union’s +Horizon 2020 research and innovation programme (grant agreement No. 947923). The research of +M. S. was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) +under Germany’s Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics– +Geometry–Structure, and the Collaborative Research Centre CRC 1442, Geometry: Deformations +and Rigidity. Part of this article was finalized while A. C. was visiting the ETH-FIM, whose support +and excellent working conditions are gratefully acknowledged. +2 Notation and standing assumptions +2.1 Boundary value problems for Schrödinger operators on Lipschitz domains +Let Ω be a Lipschitz domain of a smooth, compact d-dimensional manifold M with (possibly empty) +boundary ∂M, by which we mean here a nonempty, open subset of M whose boundary is everywhere +locally representable as the graph of a Lipschitz function. We do not require – at least in general – +Ω to be connected, and we admit the case Ω = M (where Ω denotes the closure of Ω in M), when +of course ∂Ω = ∂M, the boundary of the ambient manifold in question. Throughout this article we +will in fact assume d ≥ 2. +We are going to study the spectrum of a given Schrödinger operator on Ω subject to boundary +conditions and, sometimes, symmetry constraints. Such symmetry constraints will be encoded in +terms of equivariance with respect to a certain group action, which we shall specify at due place. +The Schrödinger operator +∆g + q +is determined by the data of a given smooth Riemannian metric g on Ω and a given smooth (i. e. +C∞) function q: Ω → R. To avoid ambiguities, we remark here that a function (or tensor field) on +4 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +Ω smooth if it is the restriction of a smooth tensor field on M or – equivalently – on a relatively +open set containing Ω. +The boundary conditions are specified by another smooth function r: Ω → R and a decomposition +∂Ω = ∂DΩ ∪ ∂NΩ ∪ ∂RΩ +(2.1) +where the sets on the right-hand side are the closures of pairwise disjoint open subsets ∂DΩ, ∂NΩ, +and ∂RΩ of ∂Ω. +Somewhat more specifically, we will consider the spectrum of the operator ∆g + q subject to the +Dirichlet, Neumann, and Robin conditions +� +� +� +� +� +� +� +u = 0 +on ∂DΩ, +du(ηΩ +g ) = 0 +on ∂NΩ, +du(ηΩ +g ) = ru +on ∂RΩ, +(2.2) +where ηΩ +g is the almost-everywhere defined outward unit normal induced by g on ∂Ω. +It is obviously the case that the Neumann boundary conditions can be regarded as a special case of +their inhomogenous counterpart, however it is convenient – somewhat artificially – to distinguish +them in view of the later applications we have in mind, to the study of the Morse index of free +boundary minimal surfaces. +2.2 Sobolev spaces and traces +To pose the problem precisely we introduce the Sobolev space H1(Ω, g) consisting of all real-valued +functions in L2(Ω, g) which have a weak g-gradient whose pointwise g-norm is also in L2(Ω, g); then +H1(Ω, g) is a Hilbert space equipped with the inner product +⟨u, v⟩H1(Ω,g) := +� +Ω +�uv + g(∇gu, ∇gv) +� dH d(g), +integrating with respect to the d-dimensional Hausdorff measure induced by g. (We say a function +u ∈ L1 +loc(Ω, g) has a weak g-gradient ∇gu if ∇gu is a measurable vector field on Ω with pointwise +g norm in L1 +loc(Ω, g) and +� +Ω g(X, ∇gu) dH d(g) = − +� +Ω u divg X dH d(g) for every smooth vector +field X on Ω of relatively compact support, where divg X is the g divergence of X; ∇gu is uniquely +defined whenever it exists, modulo vector fields vanishing almost everywhere.) +Under our assumptions on ∂Ω we have a bounded trace map H1(Ω, g) → L2(∂Ω, g), extending +the restriction map C1(Ω) → C0(∂Ω). (The Hilbert space L2(∂Ω, g) is defined using either the +(d−1)-dimensional Hausdorff measure H d−1(g) induced by g or, equivalently, the almost-everywhere +defined volume density induced by g on ∂Ω.) In fact, we have not only boundedness of this map +but also the stronger inequality +∥u|∂Ω∥L2(∂Ω,g) ≤ C(Ω, g) +� +ϵ∥u∥H1(Ω,g) + C(ϵ)∥u∥L2(Ω,g) +� +(2.3) +for all u ∈ H1(Ω, g), all ϵ > 0, some C(Ω, g) independent of u and ϵ, and some C(ϵ) independent +of u and (Ω, g). (This can be deduced, for example, by inspecting the proof of Theorem 4.6 in [9]: +5 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +specifically, we can apply the Cauchy–Schwarz inequality (weighting with ϵ, as standard) to the +inequality immediately above the line labeled (⋆ ⋆ ⋆) on page 158 of the preceding reference, whose +treatment of Lipschitz domains in Euclidean space is readily adapted to our setting.) +For each C ∈ {D, N, R}, indicating one of the boundary conditions we wish to impose, by composing +the preceding trace map with the restriction L2(∂Ω, g) → L2(∂CΩ, g) , since ∂CΩ is open in ∂Ω, we +also get a trace map ·|∂C : H1(Ω, g) → L2(∂CΩ, g). In practice we will consider traces on just ∂DΩ +and ∂RΩ. Considering the condition on ∂DΩ we will then define +H1 +∂DΩ(Ω, g) := {u ∈ H1(Ω, g) : u|∂DΩ = 0}, +that is obviously to be understood in the sense of traces, in the terms we just described, and we +remark that (2.3) also clearly holds with ∂Ω on the left-hand side replaced by ∂RΩ (or by ∂DΩ or +∂NΩ, but we have no need of the inequality in these cases). +2.3 Bilinear forms and their eigenvalues and eigenspaces +Corresponding to the above data we define the bilinear form T = T[Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ] by +T : H1 +∂DΩ(Ω, g) × H1 +∂DΩ(Ω, g) → R +(u, v) �→ +� +Ω +� +g(∇gu, ∇gv) − quv +� +dH d(g) − +� +∂RΩ +ruv dH d−1(g). +(2.4) +Then T is symmetric, bounded, and coercive as encoded in the following three equations respectively: +∀u, v ∈ H1 +∂DΩ(Ω, g) +T(u, v) = T(v, u), +∀u ∈ H1 +∂DΩ(Ω, g) +T(u, u) ≤ +�1 + C(Ω, g, q, r) +�∥u∥2 +H1(Ω,g), +(2.5) +∀u ∈ H1 +∂DΩ(Ω, g) +T(u, u) ≥ 1 +2∥u∥2 +H1(Ω,g) − C(Ω, g, q, r)∥u∥2 +L2(Ω,g), +(2.6) +where, for (2.5) and (2.6), one can take C(Ω, g, q, r) = ∥q∥C0(Ω) + C(Ω, g)∥r∥C0(∂RΩ), thanks to +the trace inequality (2.3). From these three properties and the Riesz representation theorem for +Hilbert spaces it follows that for some constant Λ = Λ(Ω, g, q, r) > 0 there exists a linear map +R: L2(Ω, g) → H1 +∂DΩ(Ω, g) such that T(Rf, v) + Λ⟨Rf, ιv⟩L2(Ω,g) = ⟨f, ιv⟩L2(Ω,g) for all functions +f ∈ L2(Ω, g) and v ∈ H1 +∂DΩ(Ω, g), where we have introduced the inclusion map ι: H1 +∂DΩ(Ω, g) → +L2(Ω, g). +(Of course, if f is smooth then standard elliptic interior regularity results ensures that u is as well +smooth on Ω and there satisfies the equation −(∆g + q − Λ)u = f in a classical pointwise sense.) +Since the inclusion H1(Ω, g) �→ L2(Ω, g) is compact (see for example Section 7 of Chapter 4 of +[37]) and of course the inclusion of the closed subspace H1 +∂DΩ(Ω, g) �→ H1(Ω, g) is bounded, the +aforementioned maps ι: H1 +∂DΩ(Ω, g) → L2(Ω, g) and the composite ιR: L2(Ω, g) → L2(Ω, g) are also +both compact operators. Furthermore, to confirm that ιR is symmetric we simply note that (by +appealing to the equation defining the operator R, with Rf1 and Rf2 in place of v) +⟨f2, ιRf1⟩L2(Ω,g) = T(Rf2, Rf1) + Λ⟨ιRf2, ιRf1⟩L2(Ω,g) += T(Rf1, Rf2) + Λ⟨ιRf1, ιRf2⟩L2(Ω,g) = ⟨f1, ιRf2⟩L2(Ω,g) +6 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +for all f1, f2 ∈ L2(Ω, g). That being clarified, to improve readability we will from now on refrain +from explicitly indicating the inclusion map ι in our equations. +With slight abuse of language, in the setting above we call λ ∈ R an eigenvalue of T if there exists +a nonzero u ∈ H1 +∂DΩ(Ω, g) such that +∀v ∈ H1 +∂DΩ(Ω, g) +T(u, v) = λ⟨u, v⟩L2(Ω,g), +(2.7) +and we call any such u an eigenfunction of T with eigenvalue λ. (We caution that the notions of +eigenfunctions and eigenvalues depend not only on T but also on the underlying metric g; for the +sake of convenience we choose to suppress the latter dependence from our notation.) +Hence, as a consequence of the key facts we presented before this definition, one can prove by well- +known arguments the existence of a discrete spectrum for the “shifted” elliptic operator (∆g +q)−Λ +subject to the very same boundary conditions (2.2). As a straightforward corollary, by accounting +for the shift, we obtain the following conclusions for T: +• the set of eigenvalues of T is discrete in R and bounded below, +• for each eigenvalue of T the corresponding eigenspace has finite dimension, +• there exists an Hilbertian basis {ej}∞ +j=1 for L2(Ω, g) consisting of eigenfunctions of T, +• and {ej}∞ +j=1 has dense span in H1 +∂DΩ(Ω, g). +(To avoid ambiguities, we remark that the phrase Hilbertian basis refers to a countable, complete +orthonormal system for the Hilbert space in question.) For each integer i ≥ 1 we write λi (T) for +the ith eigenvalue of T (listed with repetitions in nondecreasing order, in the usual fashion). There +holds the usual min-max characterization +λi (T) = min +� +max +� T(w, w) +∥w∥2 +L2(Ω,g) +: 0 ̸= w ∈ W +� +: W ⊂ +subspace +H1 +∂DΩ(Ω, g), dim W = i +� +. +(2.8) +Next, for any t ∈ R we let E=t(T) denote the (possibly trivial) linear span, in H1 +∂DΩ(Ω, g), of the +eigenfunctions of T with eigenvalue t, and, more generally, for any t ∈ R and any binary relation ∼ +on R (in practice <, ≤, >, ≥, or =) we set +E∼t(T) := ClosureL2(Ω,g) +� +Span +�� +s∼t +E=s(T) +�� +and we denote the corresponding orthogonal projection by +π∼t +T : L2(Ω, g) → E∼t(T). +That is, the space E∼t(T) has been defined to be the closure in L2(Ω, g) of the span of all +eigenfunctions of T having eigenvalue λ such that λ ∼ t. Of course E∼t(T) is a subspace of +H1 +∂DΩ(Ω, g) – in particular – whenever the former has finite dimension. Taking ∼ to be equality +clearly reproduces the originally defined space E=t(T). +For future use observe that the above spectral theorem for T implies +(E∼t(T))⊥L2(Ω,g) = E̸∼t(T), +E, ≥, +(2.9) +7 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +and +∀u ∈ H1 +∂DΩ(Ω, g) ∩ +� +E≤t(T) ∪ E≥t(T) +� +T(u, u) = t∥u∥2 +L2(Ω,g) ⇒ u ∈ E=t(T), +throughout which t is any real number (not necessarily an eigenvalue of T) and where in the first +equality of (2.9) ∼ is any relation on R and ̸∼ its negation (so that {s ̸∼ t} = R \ {s ∼ t} for any +t ∈ R). +Index and nullity. +In the setting above, and under the corresponding standing assumption, we +shall define the non-negative integers +ind(T) := dim E<0(T) +and +nul(T) := dim E=0(T), +called, respectively, the index and nullity of T. Such invariants will be of primary interest in our +applications. +2.4 Group actions +Let G be a finite group of smooth diffeomorphisms of M, each restricting to a surjective isometry of +(Ω, g). Then, as for any group of diffeomorphism of Ω, we have the standard (left) action of G on +functions on Ω via pullback: +(φ, u) �→ u ◦ φ−1 = φ−1∗u +for all φ ∈ G, u: Ω → R. +We say that a function u is G-invariant if it is invariant under this action: equivalently u ◦ φ = u for +all φ ∈ G. +We can also twist this action by orthogonal transformations on the fiber R: given in addition to G +a group homomorphism σ: G → O(1) = {−1, 1}, we define the action +(φ, u) �→ σ(φ)(u ◦ φ−1) = σ(φ)φ−1∗u +for all φ ∈ G, u: Ω → R, +and we call a function (G, σ)-invariant if it is invariant under this action. Obviously the above +standard action (φ, u) �→ u ◦ φ−1 is recovered by taking the trivial homomorphism σ ≡ 1. We +also comment that one could of course replace R by C and correspondingly O(1) by U(1) (and in +the preceding sections instead work with Sobolev spaces over C) though we restrict attention to +real-valued functions in this article. +Since, by virtue of our initial requirement, G is a group of isometries of (Ω, g), the above twisted +action yields a unitary representation of G in L2(Ω, g), i. e. a group homomorphism +�σ: G → O +�L2(Ω, g) +� +φ �→ σ(φ)φ−1∗ +(2.10) +whose target are the global isometries of L2(Ω, g); we note that the same conclusions hold true with +H1(Ω, g) in place of L2(Ω, g). The corresponding subspaces of (G, σ)-invariant functions, in L2(Ω, g) +8 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +or H1(Ω, g), are readily checked to be closed, and thus Hilbert spaces themselves. That said, we +define the orthogonal projection +πG,σ : L2(Ω, g) → L2(Ω, g) +u �→ +1 +|G| +� +φ∈G +�σ(φ)u. +(2.11) +Here |G| is the order of G, which – we recall – is assumed throughout to be finite. The image of +L2(Ω, g) under πG,σ thus consists of (G, σ)-invariant functions. +Remark 2.1. One could lift the finiteness assumption, say by allowing G to be a compact Lie group, +requiring σ to be continuous, and replacing the finite average in (2.11) with the average over G with +respect to its Haar measure (which reduces to the former for finite G). However, with a view towards +our later applications, in this article we will content ourselves with the finiteness assumption, which +allows for a lighter exposition. +Henceforth we make the additional assumptions that G globally (i. e. as sets) preserves each of ∂DΩ, +∂NΩ, and ∂RΩ, and that q and r are both G-invariant. Each element of �σ(G) then preserves also +H1 +∂DΩ(Ω, g) and the bilinear form T, and the projection πG,σ commutes with the projection π∼t +T , +for any t ∈ R and binary relation ∼ on R (as above). In particular πG,σ preserves each eigenspace +E=t(T) of T, and more generally the space +E∼t +G,σ(T) := πG,σ(E∼t(T)) +(2.12) +is a subspace of E∼t(T). +For each integer i ≥ 1 we can then define λG,σ +i +(T), the ith (G, σ)-eigenvalue of T, to be the ith +eigenvalue of T having a (G, σ)-invariant eigenfunction (by definition nonzero), counting with +multiplicity as before; equivalently one can work with spaces of (G, σ)-invariant functions and derive +the analogous conclusions as in Subsection 2.3 directly in that setting. +Remark 2.2. We explicitly note, for the sake of completeness, that under no additional assumptions +on the group G and the homomorphism σ it is possible that the space of (G, σ)-invariant functions +be finite dimensional (possibly even of dimension zero). This type of phenomenon happens, for +instance, when every point of the manifold M is a fixed point of a σ-odd isometry. In this case, +all conclusions listed above still hold true, but need to be understood with a bit of care: the +corresponding sequence of eigenvalues λG,σ +1 +(T) ≤ λG,σ +2 +(T) ≤ . . . will in fact just be a finite sequence, +consisting say of I(G, σ) elements, counted with multiplicity as usual; we shall formally convene that +λG,σ +i +(T) = +∞ for i > I(G, σ). That being said, we also remark that this phenomenon patently +does not occur for the Jacobi form of the two sequences of free boundary minimal surfaces we +examine in Sections 4 and 5. +In this equivariant framework we still have the corresponding min-max characterization +λG,σ +i +(T) = min +� +max +� T(w, w) +∥w∥2 +L2(Ω,g) +: 0 ̸= w ∈ W +� +: W ⊂ +subspace +πG,σ +�H1 +∂DΩ(Ω, g) +�, dim W = i +� +. +(2.13) +9 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +We also define the (G, σ)-index and (G, σ)-nullity +indσ +G(T) := dim E<0 +G,σ(T) +and +nulσ +G(T) := dim E=0 +G,σ(T) +of T. Obviously we can recover E∼t(T), λi (T), and the standard index and nullity by taking G to +be the trivial group. As mentioned in the introduction, we reiterate that it is one of the goals of the +present article to study, for fixed g and T, how these numbers (index indσ +G(T) and nullity nulσ +G(T)) +depend on G and σ. +Terminology. +For the sake of brevity, we shall employ the phrase admissible data to denote any +tuple (Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) satisfying all the standing assumptions presented up to now. +We digress briefly to highlight two important special cases, which warrant additional notation. +Example 2.3 (Actions of order-2 groups). When |G| = 2, there are precisely two homomorphisms +G → O(1). Considering such homomorphisms, and the corresponding (G, σ)-invariant functions, we +may define G-even or G-odd functions. Hence, we may call ind+ +G and ind− +G the G-even and G-odd +index, and likewise for the nullity. Clearly, we always have +� +� +� +ind(T) = ind+ +G(T) + ind− +G(T), +nul(T) = nul+ +G(T) + nul− +G(T). +(2.14) +Example 2.4 (Actions of self-congruences of two-sided hypersurfaces). Suppose, momentarily, that +(M, g) is isometrically embedded (as a codimension-one submanifold) in a Riemannian manifold +(N, h), that the set Ω be connected and assume further that the normal bundle of M over Ω is +trivial. Then we can pick a unit normal ν on Ω and thereby identify – as usual – sections of the +normal bundle of M|Ω with functions on Ω. With this interpretation of functions on Ω in mind and +G now a finite group of diffeomorphisms of N that map Ω onto itself (as a set), and everywhere on +Ω preserve the ambient metric h meaning that φ∗h = h for any φ ∈ G, we have a natural action +given by +(φ, u) �→ sgnν(φ)(u ◦ φ−1) +for all φ ∈ G, u: Ω → R, +where sgnν(φ) := h(φ∗ν, ν) is a constant in O(1) = {1, −1}. We shall further assume that the action +of G on Ω is faithful, meaning that only the identity element fixes Ω pointly; this assumption is +always satisfied in our applications. +In this context we continue to say that a function u: Ω → R is G-invariant if u = u ◦ φ for all φ ∈ G, +and we say rather that u is G-equivariant if u = sgnν(φ)u ◦ φ for all φ ∈ G (that is, noting the +identity sgnν(φ) = sgnν(φ−1), provided u is invariant under the sgnν-twisted G action). +Similarly, in this context, we set +indG(T) := indsgnν +G +(T) +and +nulG T := nulsgnν +G +(T), +(2.15) +which we may refer to as simply the G-equivariant index and G-equivariant nullity of T. We point +out that we are abusing notation in the above definitions in that, on the right-hand side of each, +in place of G we mean really the group, isomorphic to G by virtue of the faithfulness assumption, +obtained by restricting each element of G to Ω, and in place of sgnν we mean really the corresponding +homomorphism, well-defined by the faithfulness assumption, on this last group of isometries of Ω. +We now return to the more general assumptions on G preceding this paragraph. +10 + +2 Notation and standing assumptions +A. Carlotto, M. B. Schulz, D. Wiygul +2.5 Subdomains +Suppose that Ω1 ⊂ Ω is another Lipschitz domain of M (cf. Figure 1). We shall define +∂intΩ1 := ∂Ω1 ∩ Ω, +∂extΩ1 := ∂Ω1 \ ∂intΩ1, +∂Dint +D +Ω1 := (∂extΩ1 ∩ ∂DΩ) ∪ ∂intΩ1, +∂Nint +D +Ω1 := ∂extΩ1 ∩ ∂DΩ, +∂Dint +N +Ω1 := ∂extΩ1 ∩ ∂NΩ, +∂Nint +N +Ω1 := (∂extΩ1 ∩ ∂NΩ) ∪ ∂intΩ1, +∂Dint +R +Ω1 := ∂extΩ1 ∩ ∂RΩ, +∂Nint +R +Ω1 := ∂extΩ1 ∩ ∂RΩ. +(2.16) +In this way we prepare to pose two different sets of boundary conditions on Ω1, whereby, roughly +speaking, in both cases ∂Ω1 inherits whatever boundary condition is in effect on ∂Ω wherever the +two meet (corresponding to ∂extΩ1) and the two sets of conditions are distinguished by placing +either the Dirichlet or the Neumann condition on the remainder of the boundary (corresponding to +∂intΩ1). Naturally associated to these two sets of conditions are the bilinear forms +T Dint +Ω1 +:= T[Ω1, g, q, r, ∂Dint +D +Ω1, ∂Dint +N +Ω1, ∂Dint +R +Ω1], +T Nint +Ω1 +:= T[Ω1, g, q, r, ∂Nint +D +Ω1, ∂Nint +N +Ω1, ∂Nint +R +Ω1], +(2.17) +defined, respectively, on the Sobolev spaces H1 +∂Dint +D +Ω1(Ω1, g) and H1 +∂Nint +D +Ω1(Ω1, g). +Recalling (G, σ) from above, with the tacit understanding that (Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) is +admissible, we further assume that each element of G maps Ω1 onto itself; since G preserves Ω +and respects the decomposition (2.1), it follows that it also respects the decompositions (2.16). +Somewhat abusively, we shall write �σ and πG,σ not only for the maps (2.10) and (2.11) but also +for their counterparts with Ω replaced by Ω1, which are well-defined under our assumptions. The +spaces E∼t +G,σ(T Dint +Ω1 ) and E∼t +G,σ(T Nint +Ω1 ) as in (2.12), are then also well-defined. +∂RΩ +∂DΩ +∂DΩ +∂NΩ +∂NΩ +Ω1 +∂extΩ1 ∩ ∂NΩ +∂extΩ1 ∩ ∂DΩ +∂extΩ1 ∩ ∂RΩ +∂intΩ1 +Figure 1: Example of a Lipschitz domain Ω with subdomain Ω1. +11 + +3 Fundamental tools +A. Carlotto, M. B. Schulz, D. Wiygul +3 Fundamental tools +3.1 Index and nullity bounds in the style of Montiel and Ros +Recalling the notation and assumptions of Section 2, suppose now that we have not only Ω1 ⊂ Ω as +above, but also (open) Lipschitz subdomains Ω1, . . . , Ωn ⊂ Ω which are pairwise disjoint, each of +which satisfies the same assumptions as Ω1 in Section 2.5, and whose closures cover Ω. In particular, +we assume that each element of the group G maps each subdomain Ωi onto itself. We assume +further that G acts transitively on the connected components of Ω and note that this last condition +is always satisfied in the important special case that Ω is connected. +Proposition 3.1 (Montiel–Ros bounds on the number of eigenvalues below a threshold). With +assumptions as in the preceding paragraph and notation as in Section 2, the following inequalities +hold for any t ∈ R +(i) dim E 0 less than the injectivity radius of (M, g), say δ0, we +are given a Lipschitz domain Ωδ ⊂ Ω such that (Ωδ, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ, G, σ) are also admissible +data (with suitable restrictions of tensors and functions tacitly understood), and whose complement +Kδ := Ω \ Ωδ satisfies +� +p∈S +Bf1(δ)(p) ⊂ Kδ ⊂ +� +p∈S +Bf2(δ)(p) +(3.8) +for some finite set of points S ⊂ Ω and monotone functions f1, f2 : [0, δ0[ → R≥0 such that +limδ→0 f2(δ) = 0. Consider the sets as in (2.16) with Ωδ in lieu of Ω1 as well as the associated +bilinear form +T Dint +Ωδ +:= T[Ωδ, g, q, r, ∂Dint +D +Ωδ, ∂Dint +N +Ωδ, ∂Dint +R +Ωδ]. +Then for each integer i ≥ 1 +λG,σ +i +� +T Dint +Ωδ +� +≥ λG,σ +i +(T) , +(3.9) +and we have +lim +δ→0 λG,σ +i +� +T Dint +Ωδ +� += λG,σ +i +(T) . +(3.10) +The conclusion simply relies on the fact that points have null W 1,s-capacity in Rn for 1 ≤ s ≤ n +and so, in particular, have null W 1,2-capacity in Rn for any n ≥ 2; for the sake of completeness, +we provide a self-contained argument focusing on the case of surfaces (d = 2), where a logarithmic +cutoff trick is required, and omit the simpler modifications for d ≥ 3. +Proof. Given any uδ, vδ ∈ H1 +∂Dint +D +Ωδ(Ωδ), postulated to be (G, σ)-invariant, it is standard to note +that their extensions by 0, say uδ, vδ respectively, belong to H1 +∂DΩ(Ω), that such functions are +themselves (G, σ)-invariant, and for any δ ∈ (0, δ0) there hold ⟨uδ, vδ⟩L2(Ωδ,g) = ⟨uδ, vδ⟩L2(Ω,g) +and T Dint +Ωδ (uδ, uδ) = T(uδ, uδ). Hence, it follows at once from the variational characterization of +eigenvalues, (2.13), that for each integer i ≥ 1 we have indeed λG,σ +i +� +T Dint +Ωδ +� +≥ λG,σ +i +(T), which is +the first claim. Appealing again to the domain monotonicity, it actually suffices to check (3.10) +in the case when Kδ is in fact a union of metric balls, namely when we have equality in (3.8), for +f1 = f2. To simplify the notation we can (without loss of generality, up to reparametrization) +assume in fact f2(δ) = δ for any δ in the assumed domain. That said, given any u, v ∈ H1 +∂DΩ(Ω), +19 + +3 Fundamental tools +A. Carlotto, M. B. Schulz, D. Wiygul +(G, σ)-invariant, and δ > 0 (small as in the statement) one can simply define uδ = uϕδ, vδ = vϕδ +where (for r := dg(p, q) and p ∈ S) we set +ϕδ(q) = +� +� +� +� +� +� +� +0 +if r ≤ δ3/4 +3 − 4log r +log δ +if δ3/4 ≤ r ≤ δ1/2 +1 +otherwise. +It is then clear that uδ, vδ ∈ H1 +∂Dint +D +Ωδ(Ωδ), that such functions are (G, σ)-invariant, and, in addition, +lim +δ→0 T Dint +Ωδ (uδ, uδ) = T(u, u), lim +δ→0⟨uδ, vδ⟩L2(Ωδ,g) = ⟨u, v⟩L2(Ω,g). +Hence, again appealing to (2.13), we must conclude +lim sup +δ→0 +λG,σ +i +� +T Dint +Ωδ +� +≤ λG,σ +i +(T) . +(3.11) +whence, combining this inequality with the one above, the conclusion follows. +Corollary 3.10. Given the setting and the assumptions of Proposition 3.9, we have +lim +δ→0 indσ +G(T Dint +Ωδ ) = indσ +G(T). +3.4 Conformal change in dimension two +In this section we suppose, in addition to the assumptions above, that d = dim M = 2 and that +we are given a smooth, strictly positive, G-invariant function ρ on Ω. Note that the above bilinear +form T of (2.4) is invariant under scaling, namely under the simultaneous transformations g �→ ρ2g, +q �→ ρ−2q and r �→ ρ−1r: +T +�Ω, ρ2g, ρ−2q, ρ−1r, ∂DΩ, ∂NΩ, ∂RΩ +� = T +�Ω, g, q, r, ∂DΩ, ∂NΩ, ∂RΩ +� +with the corresponding domains H1 +∂DΩ(Ω, ρ2g) and H1 +∂DΩ(Ω, g) agreeing as sets of functions and +having equivalent norms. This claim needs a clarification: the standard H1-norms of H1 +∂DΩ(Ω, ρ2g) +and H1 +∂DΩ(Ω, g) are only equivalent up to constants that depend on the extremal (inf and sup) +values of the conformal factor ρ. +In general, the eigenvalues (as defined in Subsection 2.3) will be affected by the conformal scaling, +and yet the index and nullity are nonetheless invariant when this operation is performed: +Proposition 3.11 (Invariance of index and nullity under conformal change in dimension two). +With assumptions as in the preceding paragraph +indσ +G(T, ρ2g) = indσ +G(T, g) +and +nulσ +G(T, ρ2g) = nulσ +G(T, g). +Proof. By definition u ∈ E=0 +G,σ(T, g) if and only if u is (G, σ)-invariant and T(u, v) = 0 for all +v ∈ H1 +∂DΩ(Ω, g) (and likewise if each g is replaced by ρ2g), so the nullity equality is clear. For the +index, because we can reverse the roles of g and ρ2g by replacing ρ with ρ−1, it suffices to check that +the claim holds with ≥ in place of =. This follows at once from the min-max characterization (2.13) +applied to the (G, σ)-eigenvalues of (T, ρ2g), by considering the “competitor” subspace E<0 +G,σ(T, g) +in the minimization problem therein, for i = indσ +G(T, g). +20 + +4 Free boundary minimal surfaces in the ball: a first application +A. Carlotto, M. B. Schulz, D. Wiygul +4 Free boundary minimal surfaces in the ball: a first application +From now on, we specialize our study to the case when Ω = M is a properly embedded free +boundary minimal surface, henceforth denoted by Σ, of the closed unit ball B3 := {(x, y, z) ∈ R3 : +x2 + y2 + z2 ≤ 1} in Euclidean space (R3, gR3). Observe that, by the maximum principle, every +embedded free boundary minimal surface is properly embedded. +As anticipated in the introduction, our task here will be to obtain quantitative estimates on the +Morse index of free boundary minimal surfaces, hence our Schrödinger operator is the Jacobi (or +stability) operator on Σ acting on functions subject to the Robin condition +du(ηΣ +gR3) = u +on ∂Σ, +(4.1) +namely: q = |AΣ|2, the squared norm of the second fundamental form of Σ, and ∂DΣ = ∂NΣ = ∅, +∂RΣ = ∂Σ, r = 1. Correspondingly, as our bilinear form T we will consider the index (or stability +or Jacobi) form of Σ, which we will denote by QΣ. We define the index and nullity of Σ in the usual +way, setting +ind(Σ) := ind(QΣ) +and +nul(Σ) := nul(QΣ), +and we likewise define the G-equivariant index and nullity of Σ, indG(Σ) and nulG(Σ) in the sense +of (2.15), when given a group G < O(3) of symmetries of Σ one considers the associated sign +homomorphism. More generally, we will also study the (G, σ)-index and (G, σ)-nullity of Σ, indσ +G(Σ) +and nulσ +G(Σ), when given a group G and, further, a homomorphism σ: G → O(1) (thus, in either +case, these expressions are to be understood by replacing Σ by QΣ). +It has already been mentioned above how general lower bounds for the index, linear in the topological +data (genus and number of boundary components), have been obtained in [2], and by Sargent in +[34] in the special case when the ambient manifold is a convex body in Euclidean R3. We begin this +section by presenting an alternative lower bound (Proposition 4.2 below) in terms of symmetries, +which, though much less general in nature, nevertheless yields sharper lower bounds for many of the +known examples (in terms of the coefficients describing the linear growth rate as a function of the +topological data). Before proceeding, we pause to explain some notation we will find convenient. +Cylindrical coordinates and wedges. +We equip R3 (and so, by restriction, also B3) with standard +– up to labeling – cylindrical coordinates (r, θ, z), so that the point with cylindrical coordinates +(r0, θ0, z0) has Cartesian coordinates (x, y, z) = (r0 cos θ0, r0 sin θ0, z0). For our purposes it will be +convenient to allow arbitrary real values for θ. Given real numbers α ≤ β, we also define the closed +wedge +W β +α := {(r cos θ, r sin θ, z) : r ≥ 0, θ ∈ [α, β], z ∈ R}, +(4.2) +with the half-plane W α +α accommodated as a degenerate wedge. In particular, our convention implies +{θ = α} = W α +α ∪ W α+π +α+π . +21 + +4 Free boundary minimal surfaces in the ball: a first application +A. Carlotto, M. B. Schulz, D. Wiygul +Notation for symmetries. +Given a plane Π ⊂ R3 through the origin, we write RΠ ∈ O(3) for +reflection through Π. Similarly, given a directed line ξ ⊂ R3 through the origin and an angle θ ∈ R, +we write Rθ +ξ for rotation about ξ through angle α in the usual right-handed sense. Typically we will +be interested not exclusively in such a rotation Rθ +ξ but rather in the cyclic subgroup it generates, +with the result that it will never really be important to associate a direction to ξ. Given symmetries +T1, . . . , Tn ∈ O(3), we write ⟨T1, . . . , Tn⟩ for the subgroup they generate. +The order-2 groups generated by reflections through planes will figure repeatedly in the sequel +(beginning with the following proposition), so for succinctness of notation, given a plane Π ⊂ R3 +through the origin, we agree to set Π := ⟨RΠ⟩. In such context, consistently with the general +convention we defined above, we will employ the apex + (respectively: −) to denote functions that +are even (respectively: odd) with respect to the reflection through Π. Similarly (but less frequently), +if ξ is a line through the origin in R3, we will write ξ for the order-2 group generated by reflection +Rξ through ξ (equivalently rotation through angle π in either sense about ξ). +We also pause to name the following three subgroups of O(3), which will be realized as subgroups of +the symmetry groups of the examples we study below and which partly pertain to the statement of +the next proposition: for each integer k ≥ 1 we set +Yk := +� +R{θ=− π +2k }, R{θ= π +2k } +� +(pyramidal group of order 2k), +Pk := +� +R{θ=− π +2k }, R{θ= π +2k }, R{z=0} +� +(prismatic group of order 4k), +Ak := +� +R{θ= π +2k }, Rπ +{y=z=0} +� +(antiprismatic group of order 4k). +(4.3) +Note in particular that we have Yk = Pk ∩ Ak. +Remark 4.1. The above three groups are so named because they are the (maximal) symmetry groups +of, respectively, a right pyramid, prism, or antiprism over a regular k-gon. See e. g. Section 2 of [6] +for pictures and further details, but we caution that the above definition of the subgroup Pk differs +slightly from that given in [6]: the two subgroups are conjugate to one another via rotation through +angle π/(2k) about the z-axis. +With this terminology and notation in place, we can then proceed with the aforementioned lower +index bound, which illustrates the Montiel–Ros methodology as developed in Section 3 and is +interesting in its own right. +Proposition 4.2 (Index lower bounds under pyramidal and prismatic symmetry; cf. [7,21]). Let +Σ be a connected, embedded free boundary minimal surface in B3. Assume that Σ is not a disc +or critical catenoid, that Σ is invariant under reflection through a plane Π1, and that Σ is also +invariant under rotation through an angle α ∈ ]0, 2π[ about a line ξ ⊂ Π1. Then α is a rational +multiple of 2π, there is a largest integer k ≥ 2 such that rotation about ξ through angle 2π/k is also +a symmetry of Σ, and +(i) ind(Σ) ≥ 2k − 1, +(ii) ind− +Π1(Σ) ≥ k − 1, and +(iii) if Σ is additionally invariant under reflection through a plane Π⊥ orthogonal to ξ, then in fact +ind+ +Π⊥(Σ) ≥ 2k − 1. +22 + +4 Free boundary minimal surfaces in the ball: a first application +A. Carlotto, M. B. Schulz, D. Wiygul +Note that the symmetries assumed in the preamble of Proposition 4.2 generate, up to conjugacy in +O(3), the group Yk from (4.3), while one instead obtains (again up to conjugacy) the group Pk by +adjoining the additional symmetry assumed in item (iii). +The proof below is an abstraction and transplantation to the free boundary setting of some index +lower bounds obtained in the course of [21] and drawing on ideas from [32]. The estimates ultimately +depend on a lower bound on the number of nodal domains of a suitable Jacobi field, which was +also the basis for earlier index estimates (of complete minimal surfaces in R3 and closed minimal +surfaces in S3) established by Choe in [7]. +Proof. By excluding the discs and critical catenoids we ensure that Σ is not S1-invariant about +ξ, implying the claim on α and the existence of the rotational symmetry about ξ through angle +of the form 2π/k, as follows. First, if the cyclic subgroup generated by rotation about ξ through +angle α were not finite, then it would be dense in the SO(2) subgroup of rotations about ξ, but the +symmetry group of Σ is closed in O(3); yet, as already observed, our assumptions ensure that Σ has +no SO(2) symmetry subgroup. Thus α must be a rational multiple of 2π, as claimed. Now let β be +the least angle in ]0, 2π[ through which rotation about ξ is generated by the assumed rotational +symmetry through angle α, and let k be the least positive integer such that kβ ≥ 2π. Then rotation +through angle kβ − 2π, which lies in [0, β[, is also generated by the assumed rotational symmetry. +The presumed minimality of β then forces β = 2π/k. +By composing the assumed symmetries, it follows that Σ is also invariant under reflection through +each of the k − 1 planes Π2, . . . , Πk containing ξ and there meeting Π1 at angle an integer multiple +of π/k. Now suppose Π ∈ {Πi}k +i=1. We necessarily have Π∩Σ ̸= ∅ (for example since Π separates B3 +into two components and is a plane of symmetry for Σ, which is assumed to be connected). Because +Π is a plane of symmetry and Σ is embedded, these two surfaces must intersect either orthogonally +or tangentially, but in the latter case Σ must be a disc, which possibility we have excluded by +assumption; consequently, the intersection is orthogonal. Moreover, by the symmetries each of the +2k components W1, . . . , W2k of B3 \ �k +i=1 Πi then has nontrivial intersection Ωi := Σ ∩ Wi with Σ. +Note that the members of the family {Ωi}2k +i=1 are pairwise isometric and each is connected. (Indeed, +Σ is itself connected, so any two points in any single Ωi can be joined by some path in Σ, but this +path can leave Ωi only through the latter’s intersection with planes of symmetry, so we can always +produce a path connecting the two points that is entirely contained in Ωi, by repeated reflection and +replacement, if necessary.) Furthermore, each Ωi has Lipschitz boundary contained in S2 ∪ �k +i=1 Πi, +because the intersection of Σ with either S2 and any of the planes Π1, . . . , Πk is orthogonal (thus +transverse), and exactly k of the Ωi lie on each side of Π1. +Next, letting κξ be a choice of (scalar-valued) Jacobi field on Σ induced by the rotations about ξ +and again using the fact that Σ is not rotationally symmetric (and so, in particular, not planar +either), we conclude that κξ vanishes on Σ ∩ �k +i=1 Πi (because of the aforementioned orthogonality) +but does not vanish identically on any Ωi. As a result, imposing, for each i, the Robin condition +(4.1) on S2 ∩ ∂Ωi and the Dirichlet condition on ∂Ωi ∩ �k +i=1 Πi, the corresponding nullity of Ωi is at +least 1. An appeal to item (i) of Corollary 3.2 now completes the proof. Specifically: +• for our claim (i) we consider the partition of Σ into the 2k domains Ω1, . . . , Ω2k, and take G +to be the trivial group; +23 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +• for our claim (ii) we consider the partition of Σ into the k domains Ω1 ∪ Ω2, . . . , Ω2k−1 ∪ Ω2k +(i. e. we join pairs of adjacent domains), take G = ⟨RΠ1⟩ to be the group with two elements +(as in Example 2.3) and the homomorphism determined by σ(RΠ1) = −1 (thereby imposing +odd symmetry); +• for our claim (iii) we consider the partition of Σ into the 2k domains Ω1 . . . , Ω2k, take G = ⟨RΠ⊥⟩ +to be the group with two elements and the homomorphism determined by σ(RΠ⊥) = +1 +(thereby imposing even symmetry). +Thereby the proof is complete. +5 Effective index estimates for two sequences of examples +5.1 Review of the construction and lower index bounds +Like we have already alluded to in the introduction, in [6] two families of embedded free boundary +minimal surfaces in B3 were constructed by desingularizing (in the spirit of [17]) the configurations +−K0 ∪ K0 and −K0 ∪ B2 ∪ K0, where K0 is the intersection with B3 of a certain catenoidal annulus +having axis of symmetry {x = y = 0} and meeting ∂B3 (not orthogonally) along the equator ∂B2 +and orthogonally along one additional circle of latitude at height h > 0. +Proposition 5.1 (Existence and basic properties of K0). There exists a minimal annulus K0 +which is properly embedded in B3 and intersects the unit sphere ∂B3 exactly along the equator +∂0K0 := ∂B3 ∩ {z = 0} and orthogonally along a circle of latitude at height z = h ≈ 0.87028 which +we denote by ∂⊥K0 := ∂K0 \ ∂0K0. Moreover, K0 coincides with the surface of revolution of the +graph of r: [0, h] → ]0, 1[ given by r(ζ) = (1/a) cosh(aζ − s) for suitable a ≈ 2.3328 and s ≈ 1.4907. +Proof. The existence of K0 is proven in [6, Lemma 3.3]. For the numerical values of a, h and s we +refer to [6, Remark 3.9]. +That being said, these are (somewhat simplified) versions of the main existence results we proved in +[6]. +Theorem 5.2 (Desingularizations of −K0 ∪K0 [6]). For each sufficiently large integer n there exists +in B3 a properly embedded free boundary minimal surface Ξ−K0∪K0 +n +that has genus 0, exactly n + 2 +boundary components and is invariant under the prismatic group Pn from (4.3). Moreover Ξ−K0∪K0 +n +converges to −K0 ∪ K0 in the sense of varifolds, with unit multiplicity, and smoothly away from the +equator, as n → ∞. +Theorem 5.3 (Desingularizations of −K0 ∪ B2 ∪ K0 [6]). For each sufficiently large integer m +there exists in B3 a properly embedded free boundary minimal surface Σ−K0∪B2∪K0 +m +that has genus m, +exactly 3 boundary components and is invariant under the antiprismatic group Am+1 from (4.3). +Moreover Σ−K0∪B2∪K0 +m +converges to −K0 ∪ B2 ∪ K0 in the sense of varifolds, with unit multiplicity, +and smoothly away from the equator, as m → ∞. +24 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Proposition 5.4 (Lower bounds by symmetry on the index of the examples of [6]). There exist +n0, m0 > 0 such that we have the following index estimates for all integers n > n0 and m > m0 +ind+ +{z=0}(Ξ−K0∪K0 +n +) ≥ 2n − 1 +and +ind(Σ−K0∪B2∪K0 +m +) ≥ 2m + 1. +Proof. As stated in Theorem 5.2, Ξ−K0∪K0 +n +is invariant under the action of the prismatic group +Pn which is generated by the reflections through the vertical planes {θ = −π/(2n)} and {θ = +π/(2n)} and through the horizontal plane {z = 0}. As a composition of the first two reflections, +Pn also contains the rotation by angle 2π/n about the vertical axis ξ0 = {r = 0}. Applying +Proposition 4.2 (iii) with k = n, ξ = ξ0, Π1 = {θ = π/(2n)} and Π⊥ = {z = 0} we obtain +ind+ +{z=0}(Ξ−K0∪K0 +n +) ≥ 2n − 1. +Similarly, Theorem 5.3 states that Σ−K0∪B2∪K0 +m +is invariant under the action of the antiprismatic +group Am+1 which contains the reflection through the vertical plane {θ = π/(2(m + 1))} and also +the rotation by angle 2π/(m + 1) about the vertical axis ξ0. Applying Proposition 4.2 (i) then yields +ind(Σ−K0∪B2∪K0 +m +) ≥ 2m + 1. +In terms of topological data, the previous proposition (compared to [2]) provides a coefficient 2 +for the growth rate of the Morse index of Ξ−K0∪K0 +n +(respectively: Σ−K0∪B2∪K0 +m +) with respect to the +number of boundary components (respectively: of the genus), modulo an additive term. In fact, the +lower bound on the Morse index of Ξ−K0∪K0 +n +can be further improved via the following observation, +which pertains the odd contributions to the index instead (again with respect to reflections across +the {z = 0} plane in R3); incidentally this is also an example of application of Proposition 3.1 to a +collection of domains that are not pairwise isometric. +Proposition 5.5. There exists n0 > 0 such that we have the following index estimates for all +integers n > n0 +ind− +{z=0}(Ξ−K0∪K0 +n +) ≥ 3. +Proof. Let Π1 denote a vertical plane of symmetry, passing through the origin, of the surface +Ξ−K0∪K0 +n +(which, we recall, has prismatic symmetry Pn), let ξ be the line obtained as intersection of +such a plane with {z = 0} and let finally Π2 = ξ⊥ be the vertical plane, again passing through the +origin, that is orthogonal to Π1. Consider on Ξ−K0∪K0 +n +the function κξ = Kξ · ν where Kξ is the +Killing vector field associated to rotations around ξ (oriented either way) and ν is a choice of the +unit normal to the surface in question. Clearly, the flow of Kξ generates a curve of free boundary +minimal surfaces around Ξ−K0∪K0 +n +, hence the function κξ lies in the kernel of the Jacobi operator of +Ξ−K0∪K0 +n +and satisfies the natural Robin boundary condition along the free boundary. Furthermore, +concerning its nodal set, we first note it contains the curves Ξ−K0∪K0 +n +∩ {z = 0}, and Ξ−K0∪K0 +n +∩ Π1. +We also claim that, for any sufficiently large n, the function κξ changes sign along the connected arc +Ξ−K0∪K0 +n +∩ Π+ +2 ∩ {z ≥ z0} +(5.1) +where Π+ +2 denote either of the half-planes determined by Π1 on Π2 and z0 > 0 is any sufficiently +small value (as we are about to describe, stressing that we can choose it independently of n). Since +25 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +ξ +r +z +K0 +−K0 +h +Figure 2: Nodal domains of the function induced by rotations around the symmetry axis ξ. +one has smooth convergence of Ξ−K0∪K0 +n +to −K0 ∪ K0 as n → ∞ away from the equator, it suffices +to verify an analogous claim for K0. In fact, it then follows from an explicit calculation that the +function induced by rotations around the symmetry axis ξ (the analogue of κξ on K0) has opposite +signs on the two endpoints of the arc K0 ∩ Π+ +2 (see Figure 2, right image), and so – assuming +without loss of generality it is negative on the equatorial point – by continuity there exists z0 > 0 +such that the same function is also strictly negative at all points of K0 ∩ Π+ +2 at height z0 ∈ [0, z0]. +In particular, we can indeed choose one such value z0 ∈ (0, z0) once and for all. +Hence, appealing to the aforementioned smooth convergence, by the intermediate value theorem +there must be a point along the arc (5.1) where κξ vanishes. Now, standard results about the +structure of the nodal sets of eigenfunctions of Schrödinger operators ensure that such a zero is +not isolated, but is either a regular point of a smooth curve or a branch point out of which finitely +many smooth arcs emanate. In either case, combining all facts above we must conclude that on +Ξ−K0∪K0 +n +∩ {z ≥ 0} the function κξ has at least four nodal domains, and thus an application of +Theorem 3.1 with t = 0, G = ⟨RΠ⟩ for Π = {z = 0} and σ(RΠ) = −1 ensures the conclusion. +Remark 5.6. Note that the very same argument would lead, when applied with no equivariance +constraint at all (i. e. when G is the trivial group) to the conclusion that for any sufficiently large n +the index of Ξ−K0∪K0 +n +is bounded from below by 7, which however is a lot worse than the bound +provided by combining Proposition 5.4 with Proposition 5.5. Furthermore, we note that one can +show that the function κξ has exactly 8 nodal domains and not more, as visualized in Figure 2. +Remark 5.7. Concerning the sharpness of the estimate given in Proposition 5.5, we note that +numerical simulations of K0 with fixed lower boundary ∂0K0 and upper boundary ∂⊥K0 constraint +to the unit sphere indicate that it has in fact index equal to 3. Roughly speaking, one negative +direction comes from “pinching” the catenoidal neck and the other two negative directions correspond +to “translations” of ∂⊥K0 on the northern hemisphere. +26 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +The rest of this section is aimed at obtaining upper bounds on the Morse index of our examples, +which is a more delicate task and one that relies crucially not only on the symmetries of the +surfaces in question but also on the way they were actually constructed (which we encode in suitable +convergence results). +5.2 Equivariant index and nullity of the models +For upper bounds we will exploit the regionwise convergence of the two families to the models glued +together in their construction. Therefore we first study the index and nullity on these models. +Equivariant index and nullity of K0. +We begin with a summary of the properties of the minimal +annulus K0 we will need. Let ∂0K0 = ∂K0 ∩ {z = 0} and ∂⊥K0 = ∂K0 \ ∂0K0 be as introduced in +Proposition 5.1 so that ∂⊥K0 is the boundary component along which K0 meets the sphere ∂B3 +orthogonally. Referring to equation (2.4), we define +QK0 +N := T +� +K0, gK0, q := +��AK0��2, r := 1, ∂DK0 := ∅, ∂NK0 := ∂0K0, ∂RK0 := ∂⊥K0 +� +(where we abuse notation in that by K0 we really mean its topological interior) to be the Jacobi +form of K0 subject to the natural geometric Robin condition (4.1) on ∂⊥K0 and to the Neumann +condition on ∂0K0. Clearly, for each k ≥ 1 the pyramidal group Yk from (4.3) preserves K0 and +each of its boundary components individually. +Lemma 5.8 (Yk-equivariant index and nullity of K0). With notation as above, for each sufficiently +large integer k +indYk(QK0 +N ) = 1 +and +nulYk(QK0 +N ) = 0. +Proof. We shall start by recalling [6, Lemma 4.4], which states that when imposing the Dirichlet +condition on ∂0K0 and the Robin condition on ∂⊥K0, then the Jacobi operator acting on Yk- +equivariant functions on K0 is invertible provided that k is sufficiently large, which means that the +equivariant nullity vanishes in this case. Considering the coordinate function u = z on K0, which +is harmonic, satisfies the Dirichlet condition on ∂0K0 and the Robin condition on ∂⊥K0, it is also +evident that the equivariant index is at least 1 in this case (cf. [6, Lemma 7.2]). This implies that +when instead the Neumann condition is imposed on ∂0K0, the equivariant index is again at least 1. +Below we prove that it is exactly 1 and the equivariant nullity is exactly 0 in the Neumann case +by showing that the second eigenvalue is strictly positive. (We note here, incidentally, that this +information also proves that the equivariant index is also exactly 1 in the case that a Dirichlet +condition is imposed on ∂0K0.) +Let a, h, s > 0 and r(ζ) = (1/a) cosh(aζ − s) be as in Proposition 5.1. In particular, we have +(r′)2 + 1 = cosh2(aζ − s). Thus, when K0 is parametrized as a surface of revolution in terms of the +coordinates (θ, ζ) with profile function r(ζ), the metric gK0 and the squared norm of the second +fundamental form AK0 on K0 are given by +gK0 = +�(r′)2 + 1 +� dζ2 + r2 dθ2, +|AK0|2 = +(−r′′)2 +((r′)2 + 1)3 + +1 +((r′)2 + 1)2r2 = +a2 + a−2 +cosh4(aζ − s). +27 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +The outward unit conormal along ∂⊥K0 = K0 ∩ {ζ = h} is given by +ηK0 = +1 +� +(r′)2(h) + 1∂ζ = +1 +cosh(ah − s)∂ζ = +1 +ar(h)∂ζ. +Assume, for the sake of a contradiction, that λ2 = λYk,sgn +2 +≤ 0, where we are considering the +spectrum of the Jacobi operator of K0 acting on Yk-equivariant functions (cf. Example 2.4), and +subject to the boundary conditions described above. Then, by first invoking the Courant nodal +domain theorem as in the proof of [6, Lemma 4.4] we may assume that the associated eigenfunction +u2 is rotationally symmetric provided that k is sufficiently large, i. e. u2 only depends on ζ and not +on θ. +That said, let u be a function on K0 which is rotationally symmetric, i. e. constant in θ. Then +∆K0u = +1 +cosh2(aζ − s) +∂2u +∂ζ2 , +and we shall consider the Jacobi operator J = ∆K0 + |AK0|2 and the eigenvalue problem +� +� +� +� +� +� +� +Ju = −λu +u′(0, ·) = 0 +(Neumann condition on ∂0K0) +u′(h, ·) = cosh(ah − s) u(h, ·) +(Robin condition on ∂⊥K0) +Since u2 must change sign, there exists z0 ∈ ]0, h[ such that u2(z0) = 0. Multiplying the eigenvalue +equation +∂2u2 +∂ζ2 + +a2 + a−2 +cosh2(aζ − s)u2 = −λ2u2 cosh2(aζ − s) +(5.2) +with u2 and integrating from ζ = 0 to ζ = z0, we obtain +� z0 +0 +−λ2u2 +2 cosh2(aζ − s) dζ = − +� z0 +0 +|u′ +2|2 dζ + +� z0 +0 +a2 + a−2 +cosh2(aζ − s)u2 +2 dζ. +Since u(z0) = 0, we can obtain the Poincaré-type inequality +� z0 +0 +|u2(ζ)|2 dζ = +� z0 +0 +��� +� ζ +z0 +u′ +2(t) dt +��� +2 +dζ ≤ +� z0 +0 +(z0 − ζ) +� z0 +ζ +|u′ +2(t)|2 dt dζ ≤ z2 +0 +2 +� z0 +0 +|u′ +2(ζ)|2 dζ. +Hence, +� z0 +0 +−λ2u2 +2 cosh2(aζ − s) dζ ≤ +� z0 +0 +� +a2 + a−2 +cosh2(aζ − s) − 2 +z2 +0 +� +u2 +2 dζ. +The right-hand side is negative if +z0 < +� +2 +a2 + a−2 ≈ 0.5962 +and so, in this case, we conclude λ2 > 0, a contradiction. +28 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Integrating the eigenvalue equation (5.2) instead from ζ = z0 to ζ = h and recalling the Robin +condition u′(h) = cosh(ah − s)u(h) along ∂⊥K0 we obtain the alternative estimate +� h +z0 +−λ2u2 +2 cosh2(aζ − s) dζ = |u(h)|2 cosh(ah − s) − +� h +z0 +|u′ +2|2 dζ + +� h +z0 +a2 + a−2 +cosh2(aζ − s)u2 +2 dζ +≤ +� +(h − z0) cosh(ah − s) − 1 +� � h +z0 +|u′ +2|2 dζ + +� h +z0 +a2 + a−2 +cosh2(aζ − s)u2 +2 dζ +≤ +� +a2 + a−2 + +2 +(h − z0)2 +� +(h − z0) cosh(ah − s) − 1 +�� � h +z0 +u2 +2 dζ +provided that (h − z0) cosh(ah − s) − 1 < 0. Now the right-hand side is negative if z0 > 0.4443. +Since the intervals [0, 0.5962] and [0.4443, h] intersect, we anyway obtain a contradiction. Thus, we +confirm the claim λ2 > 0, as desired. +Observing (as we have already done in the previous proof) that any eigenfunction “generating” the +index in Lemma 5.8 is rotationally invariant, we have the following obvious corollary (which in fact +can conversely be used to prove the lemma, with the aid of Proposition 3.8). In the statement GK0 +denotes the subgroup of O(3) preserving K0. Note that GK0 consists of rotations about the z-axis +and reflections through planes containing the z-axis. In particular GK0 is isomorphic to O(2), and +each element of GK0 preserves either choice of unit normal of K0. +Corollary 5.9 (Fully equivariant index and nullity of K0). With notation as above and recalling +the comments immediately preceding Proposition 3.8, there holds +indGK0(QK0 +N ) = 1 +and +nulGK0(QK0 +N ) = 0. +Equivariant index and nullity of B2. +The analysis for the flat disc B2 (featured in the construction +of just one of the families) is trivial, and the conclusions are as follows; in the statement we write QB2 +N +for the index form of B2 as a minimal surface with boundary in (R3, gR3) subject to the Neumann +boundary condition, namely +QB2 +N := T +� +B2, gB2, 0, 0, ∅, ∂NB2 := ∂B2, ∅ +� +. +Lemma 5.10 ((Am+1-equivariant) index and nullity of B2). With notation as above, +ind(QB2 +N ) = 0 +and +nul(QB2 +N ) = 1. +Moreover, for each integer m ≥ 0 the antiprismatic group Am+1 preserves B2 and +indAm+1(QB2 +N ) = nulAm+1(QB2 +N ) = 0. +Proof. The first line of equalities is clear, since the Jacobi operator on B2 is simply the standard +Laplacian, whose Neumann kernel is spanned by the constants (to rule out index one can just appeal +to the Hopf boundary point lemma). The invariance of B2 under each Am+1 is obvious, and the +proof is then completed by the observation that the constants are not Am+1-equivariant (for any +m ≥ 0). +29 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +From Proposition 5.10 we immediately obtain, analogously to Corollary 5.9 from Proposition 5.8, the +following corollary. In the statement O(2) refers to the group of intrinsic isometries of B2 (extended +to isometries of R2), rather than to some subgroup of O(3), and we write 1 and det for respectively +the trivial and determinant homomorphisms O(2) → O(1). The (O(2), 1)-invariant functions on B2 +are thus the radial functions, while the space of (O(2), det)-invariant functions is trivial. +Corollary 5.11 (Indices and nullities of B2 under O(2) actions). With notation as above we have +ind1 +O(2)(QB2 +N ) = 0, +nul1 +O(2)(QB2 +N ) = 1, +inddet +O(2)(QB2 +N ) = nuldet +O(2)(QB2 +N ) = 0. +Equivariant index and nullity of MΞ and MΣ. +We recall how, away from the equator S1, the +surfaces Ξ−K0∪K0 +n +and Σ−K0∪B2∪K0 +m +are constructed as graphs over (subsets of) −K0 ∪ K0 and +−K0 ∪ B2 ∪ K0. In the vicinity of S1 the surfaces are instead modeled on certain singly periodic +minimal surfaces that belong to a family discovered by Karcher [23] and generalize the classical +singly periodic minimal surfaces of Scherk [35]. We now summarize the key properties of such +models, to the extent needed later. +Proposition 5.12 (Desingularizing models). There exist in R3 complete, connected, properly +embedded minimal surfaces MΞ and MΣ having the following properties, which uniquely determine +the surfaces up to congruence: +(i) MΞ and MΣ are periodic in the y direction with period 2π and the corresponding quotient +surfaces have genus zero. +(ii) MΞ and MΣ are invariant under R{x=0}, R{y=π/2}, and R{y=−π/2}. +(iii) MΞ is invariant under R{z=0} and MΣ under R{y=z=0}. +(iv) MΞ has four ends and MΣ has six ends, all asymptotically planar. +(v) Each of MΞ and MΣ has an end contained in {x ≤ 0} ∩ {z ≥ 0} whose asymptotic plane +intersects {z = 0} at the same angle ω0 > 0 at which K0 intersects B2, and MΣ has additionally +{z = 0} as an asymptotic plane. +(vi) MΞ +fb := MΞ ∩ {x ≤ 0} ∩ {|y| ≤ π/2} and MΣ +fb := MΣ ∩ {x ≤ 0} ∩ {|y| ≤ π/2} are connected +free boundary minimal surfaces in the half slab {x ≤ 0} ∩ {|y| ≤ π/2}, with MΞ +fb invariant +under R{z=0} and MΣ +fb invariant under R{y=z=0} (cf. Figure 3). +(vii) Each of MΞ +fb \ {z = 0} and MΣ +fb \ {y = z = 0} has exactly two connected components. +(viii) MΞ has no umbilics, while the set of umbilic points of MΣ is {(0, nπ, 0) : n ∈ Z}. +(ix) The Gauss map νΞ of MΞ restricted to the closure of either component of MΞ +fb \ {z = 0} is +a bijection onto a solid spherical triangle with all sides geodesic segments of length π/2 (in +other words: a quarter hemisphere), less a point in the interior of one side. +(x) The Gauss map νΣ of MΣ restricted to the closure of either component of MΣ +fb \ {y = z = 0} +is a bijection onto a spherical lune of dihedral angle π/2 (in other words: a half hemisphere), +less one vertex and a point in the interior of one side. +30 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +x +y +z +ω0 +π +y +z +x +ω0 +π +Figure 3: The minimal surfaces MΞ +fb (left) and MΣ +fb (right) as defined in Proposition 5.12 (vi). +We refer the reader to Section 3 and Appendix A of [6] for further details and a fine analysis of +the properties of both surfaces in question. The free boundary minimal surfaces MΞ +fb and MΣ +fb are +visualized in Figure 3. +Next, we want to examine the index and nullity of MΞ +fb and MΣ +fb as free boundary minimal surfaces +in the half slab {x ≤ 0} ∩ {|y| ≤ π/2}. Because the boundary of such a domain is piecewise planar, +the corresponding Robin condition associated with the index forms of these surfaces is in fact +homogeneous (Neumann). +Let us prove an ancillary result. We will observe (in the proof of Lemma 5.16, to follow shortly) that +by virtue of the behavior of the Gauss maps described in Proposition 5.12 the analysis of the index +and nullity of MΞ +fb and MΣ +fb reduces to the following index and nullity computations for boundary +value problems on suitable Lipschitz domains of S2. +Lemma 5.13 (Index and nullity of ∆gS2 + 2 on images of Gauss maps of MΞ +fb and MΣ +fb). Set +ΩΞ +S2 := S2 ∩ {x > 0} ∩ {y > 0} ∩ {z > 0}, +ΩΣ +S2 := S2 ∩ {x > 0} ∩ {y > 0}. +Then we have the following indices and nullities, where the final row holds for any ζ ∈ ]−1, 1[ and, +throughout, T is the bilinear form (2.4) with Ω as indicated, g = gS2 the round metric, q = 2 (so +associated to the Schrödinger operator ∆gS2 + 2), ∂RΩ = ∅, ∂DΩ as indicated, and ∂NΩ = ∂Ω \ ∂DΩ: +Ω +∂DΩ +ind(T) +nul(T) +ΩΞ +S2 +∅ +1 +0 +{z = 0} +0 +1 +ΩΣ +S2 +∅ +1 +1 +{x = 0} +0 +1 +{x = 0} ∩ {z > ζ} +1 +0 +(5.3) +31 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +s +s +s +s +τ (1) +τ (4) +τ (3) +τ (2) +x +z +R1 +W1 +W1(s) +W2 +W2(s) +W3 +W3(s) +W4 +W4(s) +C +ω0 +Figure 4: A view of MΞ(s). +Proof. By Lemma 3.5 we can fill in the first four rows by identifying the index and nullity of +∆gS2 + 2 on the entire sphere subject to appropriate symmetries, the relevant spherical harmonics +being simply the restrictions of affine functions on R3. Lemma 3.5 is not directly applicable to +the final row, but by the min-max characterization (2.13) of eigenvalues the ith eigenvalue for the +bilinear form specified in the that row must lie between the ith eigenvalues of the forms specified in +the two preceding rows (≥ that of the third row and ≤ that of the fourth); moreover, the unique +continuation principle implies that both inequalities must be strict (> and <). The entries of the +final row now follow, concluding the proof. +We shall fix components of MΞ +fb \ {z = 0} and MΣ +fb \ {y = z = 0} and write ΩΞ and ΩΣ for their +respective interiors: it follows from Proposition 5.12 that νΞ|ΩΞ and νΣ|ΩΣ are diffeomorphisms +onto their images, which we can and will identify with, respectively, the triangle ΩΞ +S2 and lune ΩΣ +S2 +of Lemma 5.13, and in particular +{x = 0} ∩ ∂ΩΞ +S2 = νΞ({x = 0} ∩ ∂ΩΞ), +{y = 0} ∩ ∂ΩΞ +S2 = νΞ({y = ±π/2} ∩ ∂ΩΞ), +{z = 0} ∩ ∂ΩΞ +S2 = νΞ({z = 0} ∩ ∂ΩΞ), +and +{x = 0} ∩ ∂ΩΣ +S2 = νΣ(({x = 0} ∪ {y = z = 0}) ∩ ∂ΩΣ), +{y = 0} ∩ ∂ΩΣ +S2 = νΣ({y = ±π/2} ∩ ∂ΩΣ). +In what follows, recalling e. g. that the index of a minimal surface, when finite, can be computed by +exhaustion (cf. [10]) we conveniently introduce this notation, which pertains certain truncations of +MΞ, MΣ, MΞ +fb, and MΣ +fb. To do so, we first fix R1 > 0 large enough such that MΞ \ {x2 + z2 = R2 +1} +consists of five connected components, one component C in {y2 + z2 < R2 +1} and four components +32 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +W1, W2, W3, W4 in the complement, each of which is a graph over (a subset of) an asymptotic half +plane (see Figure 4). For each Wi let τ (i) be a unit vector parallel to the asymptotic half plane of +Wi, perpendicular to the y-axis (the axis of periodicity), and directed away from ∂Wi toward the +corresponding end, namely (up to relabeling) +τ (1) = (cos ω0, 0, sin ω0) = −τ (3), +τ (2) = (− cos ω0, 0, sin ω0) = −τ (4), +where we recall that ω0 > 0 is the angle at which K0 intersects B2. Now, given s > R1, we define +the truncations +Wi(s) := Wi ∩ {τ (i) · (x, y, z) ≤ s}, +MΞ(s) := C ∪ +4� +i=1 +Wi(s), +MΣ(s) analogously (for six ends), +MΞ +−(s) := MΞ(s) ∩ {x ≤ 0}, +MΣ +−(s) := MΣ(s) ∩ {x ≤ 0}, +(5.4) +MΞ +fb(s) := MΞ(s) ∩ MΞ +fb, +MΣ +fb(s) := MΣ(s) ∩ MΣ +fb. +For each ϵ, ϵ′ > 0 we then set +ΩΞ(ϵ) := ΩΞ ∩ MΞ +fb(ϵ−1), +ΩΣ(ϵ, ϵ′) := ΩΣ ∩ MΣ +fb(ϵ−1) ∩ {x2 + y2 + z2 > ϵ′}, +truncating ΩΞ and ΩΣ at (affine) distance ϵ−1 and excising from ΩΣ a disc with radius +√ +ϵ′ and +center at the umbilic (0, 0, 0); similarly MΣ +fb(ϵ−1, ϵ′) := MΣ +fb(ϵ−1) ∩ {x2 + y2 + z2 > ϵ′}. We then +in turn set ΩΞ +S2(ϵ) := νΞ�ΩΞ(ϵ) +� ⊂ ΩΞ +S2 as well as ΩΣ +S2(ϵ, ϵ′) := νΣ�ΩΣ(ϵ, ϵ′) +� ⊂ ΩΣ +S2. As a direct +consequence of Lemma 5.13 and Proposition 3.9 we get what follows. +Corollary 5.14. In the setting above, consider for any ϵ, ϵ′ > 0 the Schrödinger operator ∆gS2 + 2 +on the domains given, respectively, by ΩΞ +S2(ϵ) and ΩΣ +S2(ϵ, ϵ′) and subject to any of the boundary +conditions specified in the table (5.3), where the boundary is contained, respectively, in ∂ΩΣ +S2 and +∂ΩΞ +S2 and subject to Dirichlet conditions elsewhere. In other words, let T Ξ be either bilinear form +corresponding to the top two rows of (5.3), let T Σ be any bilinear form corresponding to the bottom +three rows of (5.3), and consider also the bilinear forms +T Ξ +ϵ := (T Ξ)Dint +ΩΞ +S2(ϵ) = T +� +ΩΞ +S2(ϵ), gS2, 2, 0, ∂DΩΞ +S2 ∪ (∂ΩΞ +S2(ϵ) \ ∂ΩΞ +S2), ∂NΩΞ +S2, ∅ +� +T Σ +ϵ,ϵ′ := (T Σ)Dint +ΩΣ +S2(ϵ,ϵ′) = T +� +ΩΣ +S2(ϵ, ϵ′), gS2, 2, 0, ∂DΩΣ +S2 ∪ (∂ΩΣ +S2(ϵ, ϵ′) \ ∂ΩΣ +S2), ∂NΩΣ +S2, ∅ +� +using the notation (2.17). Then there exists ϵ0 > 0 such that for all 0 < ϵ, ϵ′ < ϵ0 +ind(T Ξ +ϵ ) = ind(T Ξ) +and +ind(T Σ +ϵ,ϵ′) = ind(T Σ). +In particular, we can derive these geometric conclusions: +Corollary 5.15 (Index of MΞ +fb and MΣ +fb). We have the following even and odd indices for MΞ +fb and +MΣ +fb. +S +G +ind+ +G(S) +ind− +G(S) +MΞ +fb +{z = 0} +1 +0 +MΣ +fb +{y = z = 0} +1 +1 +33 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Proof. We will verify (as a sample) the even index asserted in the second row of the table; the other +claims are checked in the same fashion. The Gauss map of a minimal surface in R3 is (anti)conformal +away from its umbilics, with conformal factor (one half of) the pointwise square of the norm of +its second fundamental form, so by Proposition 3.11, for each ϵ, ϵ′ > 0, the index of ΩΣ +S2(ϵ, ϵ′) with +the foregoing boundary conditions (as in Corollary 5.14, according to the third row of the table +in Lemma 5.13) agrees also with the index of ΩΣ(ϵ, ϵ′) subject to the corresponding boundary +conditions. By Lemma 3.5 this last index agrees with the {y = z = 0}-even index of MΣ +fb(ϵ−1, ϵ′) +subject to the Dirichlet condition along the excisions and the Neumann condition everywhere else. +Hence, thanks to Corollary 5.14, such a value of the index is equal to 1 for any sufficiently small +ϵ, ϵ′. We now conclude, first letting ϵ′ → 0 and appealing to Proposition 3.9 to control the effect of +the excision near (0, 0, 0), and then appealing to the aforementioned characterization of the Morse +index via exhaustions, that MΣ +fb indeed has {y = z = 0}-index 1. +For use in the following subsection we fix a smooth cutoff function Ψ: [0, ∞[ → [0, 1] that is +constantly 1 on {x ≤ 1} and constantly 0 on {x ≥ 2}, and we define on MΞ and MΣ the functions +and metrics +ψΞ := (Ψ ◦ |x|)|MΞ, +ρΞ := +� +ψΞ + 1 +2 +���AMΞ��� +2 +(1 − ψΞ), +hΞ := (ρΞ)2gMΞ, +ψΣ := (Ψ ◦ |x|)|MΣ, +ρΣ := +� +ψΣ + 1 +2 +���AMΣ��� +2 +(1 − ψΣ), +hΣ := (ρΣ)2gMΣ. +(5.5) +Note that ρΞ is invariant under R{z=0}, ρΣ under R{y=z=0}, and both are invariant under R{x=0}, +R{y=−π/2}, and R{y=π/2}. It is natural to associate to MΞ +fb, regarded as a free boundary minimal +surface in the slab {x ≤ 0} ∩ {|y| ≤ π/2}, the stability form QMΞ +fb, defined at least on smooth +functions of compact support by +QMΞ +fb(u, v) := +� +MΞ +fb +gMΞ�∇gMΞu, ∇gMΞv +� dH 2(gMΞ) − +� +MΞ +fb +���AMΞ��� +2 +gMΞuv dH 2(gMΞ). +From the identity +QMΞ +fb(u, v) = +� +MΞ +fb +hΞ�∇hΞu, ∇hΞv +� dH 2(hΞ) − +� +MΞ +fb +���AMΞ��� +2 +hΞuv dH 2(hΞ) +and the manifest boundedness of +��AMΞ��2 +hΞ = (ρΞ)−2��AMΞ��2 +gMΞ we see that QMΞ +fb is in fact well-defined +on H1(MΞ +fb, hΞ). Likewise, the analogously defined QMΣ +fb is well-defined on H1(MΣ +fb, hΣ). +We now point out that we can identify the interiors of MΞ +fb and MΣ +fb under respectively the metrics +hΞ and hΣ as Lipschitz domains as in the setting of Section 2. Concretely, we first consider the +Riemannian quotients � +MΞ and � +MΣ of (MΞ, hΞ) and (MΣ, hΣ) under a fundamental period. Then +� +MΞ is diffeomorphic to S2 with four points removed and � +MΣ to S2 with six points removed. By +virtue of (5.5) and the behavior of the Gauss maps as outlined in Proposition 5.12, we can in fact +choose the last two diffeomorphisms so that they are isometries on neighborhoods of the punctures. +In this way we obtain smooth Riemannian compactifications. By composing the defining projection +of each tower onto its quotient by a fundamental period with the corresponding embedding into the +compactification we identify (via isometric embedding) the interior of MΞ +fb under hΞ and the interior +34 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +of MΣ +fb under hΣ with Lipschitz domains � +MΞ +fb and � +MΣ +fb in the two respective compactifications, and +we likewise identify ∂MΞ +fb and ∂MΣ +fb with subsets of ∂� +MΞ +fb and ∂� +MΣ +fb respectively. Of course, the role +of the “ambient manifold” for such Lipschitz domains is played respectively by the Riemannian +manifolds (S2, hΞ) and (S2, hΣ); here, with slight abuse of notation, we have tacitly extended the +metrics in question across the four and six punctures respectively. +Next, recalling the definition of T from (2.4), we define the bilinear form +Q� +MΞ +fb := T +� +� +MΞ +fb, hΞ, q = (ρΞ)−2���AMΞ��� +2 +gMΞ, r = 0, ∂D� +MΞ +fb = ∅, ∂N� +MΞ +fb = ∂� +MΞ +fb, ∂R� +MΞ +fb = ∅ +� +, +where (as we shall do generally in the sequel for functions defined on MΞ or MΣ, without further +comment) for the potential we tacitly interpret the right-hand side as a function on � +MΞ +fb; we define +Q� +MΣ +fb in analogous fashion. We then have (cf. Section 3.4) the equalities +Q� +MΞ +fb = QMΞ +fb on H1(MΞ +fb, hΞ) +and +Q� +MΣ +fb = QMΣ +fb on H1(MΣ +fb, hΣ). +(5.6) +Lemma 5.16 (Index and nullity of Q� +MΞ +fb and Q� +MΣ +fb). With definitions as in the preceding paragraph +we have the following indices and nullities. +S +G +ind+ +G(QS) +nul+ +G(QS) +ind− +G(QS) +nul− +G(QS) +� +MΞ +fb +{z = 0} +1 +0 +0 +1 +� +MΣ +fb +{y = z = 0} +1 +1 +1 +0 +Proof. The first row follows from a direct application of Proposition 3.11 in conjunction with the +first two rows of the table in Lemma 5.13. Indeed, in this case there are no umbilic points in play +(for, recall, MΞ has no umbilic points) and the Gauss map furnishes an (anti)conformal map from the +compactified quotient onto S2. For MΣ, however, the corresponding conformal factor degenerates at +the umbilic at (0, 0, 0), as all of its translates. Nevertheless, aided by Lemma 3.5 and Corollary 3.10 +we can verify the indices in the second row in much the same fashion, applying Proposition 3.11 on +suitable subdomains (obtained by removing smaller and smaller neighborhoods of the origin). +For the nullities, however, we employ an ad hoc argument, since one cannot expect an analogue +of the aforementioned Corollary 3.10 to hold true in general. That said, we observe first that the +translations in the z direction induce a nontrivial, smooth, bounded, ({y = z = 0}, +)-invariant +(scalar-valued) Jacobi field on MΣ which readily implies it to define an element of H1(MΣ +fb, hΣ). This +shows, in view of (5.6), that the nullities in question are at least the values indicated in the table. +On the other hand, (appealing to Lemma 3.5 for the regularity) each element, say u: � +MΣ +fb → R, of +the eigenspace with eigenvalue zero corresponding to the nullities in question is smooth and bounded. +If we restrict it to ΩΣ ⊂ � +MΣ +fb and consider the precomposition with the inverse of the Gauss map +(which, let us recall, yields an (anti)conformal diffeomorphism νMΣ : ΩΣ → ΩΣ +S2), then the resulting +function u0 := u◦(νMΣ)−1 satisfies (∆gS2 +2)u0 = 0 and so we get an element contributing to nul(T) +where T is as encoded in the third (respectively: the fifth) row of the table (5.3) when starting from +the ({y = z = 0}, +)-invariant (respectively: the ({y = z = 0}, −)-invariant) problem on � +MΣ +fb. +It is clear that one thereby gets injective maps of vector spaces, and so from Lemma 5.13 +nul+ +G(Q� +MΣ +fb) ≤ 1, +nul− +G(Q� +MΣ +fb) ≤ 0 +35 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +which in particular implies that such maps are, a posteriori, linear isomorphisms, and thus completes +the proof. +When we wish to consider the sets MΞ +fb(s) and MΣ +fb(s) endowed respectively with the metrics hΞ +and hΣ, we shall denote them by � +MΞ +fb(s) and � +MΣ +fb(s). Recalling the notation of Subsection 2.5, we +further define +Q� +MΞ +fb(s) +D +:= +� +Q� +MΞ +fb +�Dint +� +MΞ +fb(s) +and +Q� +MΞ +fb(s) +N +:= +� +Q� +MΞ +fb +�Nint +� +MΞ +fb(s). +(5.7) +In short, we are adjoining respectively Dirichlet or Neumann boundary conditions along the cuts. +Lemma 5.17 (Spectra of Q� +MΞ +fb(s) and Q� +MΣ +fb(s)). For each integer i ≥ 1 +lim +s→∞ λ{z=0},± +i +� +Q� +MΞ +fb(s) +D +� += lim +s→∞ λ{z=0},± +i +� +Q� +MΞ +fb(s) +N +� += λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim +s→∞ λ{y=z=0},± +i +� +Q� +MΣ +fb(s) +D +� += lim +s→∞ λ{y=z=0},± +i +� +Q� +MΣ +fb(s) +N +� += λ{y=z=0},± +i +� +Q� +MΣ +fb +� +, +for any consistent choice of + or − on both sides of each equality. +Proof. We will write down the proof of the two equalities in the first line for the + choice, as the +remaining cases can be proved in the same way. First note that Proposition 3.9 gives us +lim +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +D +� += λ{z=0},+ +i +� +Q� +MΞ +fb +� +. +Using the min-max characterization (2.13) of eigenvalues we then also get +lim sup +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +N +� +≤ lim sup +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +D +� += λ{z=0},+ +i +� +Q� +MΞ +fb +� +. +The key step now in establishing +lim inf +s→∞ λ{z=0},+ +i +� +Q� +MΞ +fb(s) +N +� +≥ λ{z=0},+ +i +� +Q� +MΞ +fb +� +(which completes the proof) is to construct a family of (appropriately symmetric) linear extension +operators Es : H1(� +MΞ +fb(s)) → H1(� +MΞ +fb) uniformly bounded in s, assuming s ≥ s0 for some universal +s0 > 0. With these extensions in hand it is straightforward, for example, to adapt the argument for +(3.6) in the proof of Proposition 3.8. +We now construct the Es extension operators. By the imposed symmetry (in the case under +discussion even reflection through {z = 0}) and by taking s large enough, it suffices to specify the +extension on a single end W, a graph over a subset of the corresponding asymptotic plane Π (with +τ the corresponding defining vector, recalling the notation preceding (5.4)). Let ϖ: W → Π be +the associated projection. By partitioning the given function using appropriately chosen smooth +cutoff functions (fixed independently of s), it in fact suffices to consider the extension problem for +a function v ∈ H1(W ∩ MΞ +fb(s), hΞ) such that the support of ϖ∗v is compactly contained in the +rectangle (expressed in the notation of (5.4)) +{0 < τ · (x, y, z) ≤ s} ∩ {−π ≤ 2y ≤ π}. +36 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +We can extend ϖ∗v via even reflection through the s side of the above rectangle, thereby obtaining an +extension of v to an element of H1(W, hΞ). The asymptotic convergence of W to Π, the monotonic +decay of ρΞ along W toward ∞, and the conformal invariance (in the current two-dimensional +setting) of the Dirichlet energy ensure that this extension has the desired properties. +5.3 Deconstruction of the surfaces and regionwise geometric convergence +We first take a moment to briefly review the constructions of the surfaces from [6]. First (cf. +[6, Section 3]), an approximate minimal surface in B3, called the initial surface, whose boundary +is contained in ∂B3 and which meets ∂B3 exactly orthogonally, is fashioned by hand, via suitable +interpolations, from the models (K0, MΞ or MΣ, and for Σ−K0∪B2∪K0 +m +also B2). Second (cf. [6, +Section 5]), the final exact solution is identified as the normal graph of a small function over +the approximate solution. For what pertains this second step we wish only to highlight that the +assignment of graph to function is made using not the usual Euclidean metric gR3 but instead +an O(3)-invariant metric (fixed once and for all, independently of the data n or m) conformally +Euclidean, and called the auxiliary metric. On a neighborhood of the origin this metric agrees +exactly with the Euclidean one, while on a neighborhood of ∂B3 = S2 it agrees exactly with +the cylindrical metric on S2 × R; this last property and the orthogonality of the intersection of +the initial surface with ∂B3 ensure that the boundary of the resulting graph is also in ∂B3. We +will write �Ξ−K0∪K0 +n +and �Σ−K0∪B2∪K0 +m +for the initial surfaces and ϖΞ +n : Ξ−K0∪K0 +n +→ �Ξ−K0∪K0 +n +and +ϖΣ +m : Σ−K0∪B2∪K0 +m +→ �Σ−K0∪B2∪K0 +m +for the nearest-point projections under the above auxiliary metric. +Turning to the first step, actually (because of the presence of a cokernel) one constructs for each +given n or m not just a single initial surface but a (continuous) one-parameter family of them. In +the construction this parameter is treated as an unknown and is determined only in the second +step, simultaneously with the defining function for the final surface. Here, however, we can take the +construction for granted and accordingly speak of a single initial surface, whose defining parameter +value is some definite (though not explicit) function of n or m as appropriate. Nevertheless we must +explain that this parameter enters the construction at the level of the building blocks, except for +B2, which is unaffected, as follows. First, the catenoidal annulus K0 is just one in a family Kϵ (cf. +the beginning of Subsection 3.1 in [6]) of such annuli, all rotationally symmetric about the z-axis, +depending smoothly on ϵ. The details are not critical here, but each Kϵ is the intersection with B3 +of a complete catenoid with axis the z-axis, and Kϵ meets S2 at two circles of latitude, the upper +one a circle of orthogonal intersection and the lower one the circle at height z = ϵ. Similarly, from +MΞ and MΣ we define, by explicit graphical deformation, families which here we will call MΞ +δ and +MΣ +δ (cf. the beginning of Subsection 3.2 of [6]). These deformations are the identity on the “cores” +of MΞ and MΣ and smoothly transition to translations on the ends, in the z-direction, up or down +depending on the end, and through a displacement determined by δ. Importantly, all the MΞ +δ and +MΣ +δ have the same symmetries as MΞ and MΣ respectively. Now the datum n determines building +blocks MΞ +δΞ(n) and KϵΞ(n), while the datum m determines building blocks MΣ +δΣ(m), KϵΣ(m), and B2. +We next define maps ΦΞ +n and ΦΣ +m ([6, (3.37)]) from neighborhoods of +1 +nMΞ +δΞ(n) ∩ {x ≤ 0} and +1 +m+1MΣ +δΣ(m) ∩ {x ≤ 0} respectively into B3, so as to “wrap” the cores of these surfaces around +the equator S1 approximately isometrically but to take their asymptotic half planes (in {x ≤ 0}) +onto ±KϵΞ(n) in the first case and onto ±KϵΣ(m) and B2 in the second. Thus, just referring to the +family Ξ−K0∪K0 +n +for the sake of brevity, we truncate 1 +nMΞ +δΞ(n) by intersecting with {x ≥ −n3/4} and +37 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +then apply ΦΞ +n. The image is embedded (for n large enough) and contained in the ball, in fact +contained in a tubular neighborhood of S1 with radius of order n−1/4. Near the two truncation +boundary components the surface is a small graph over either ±KϵΞ(n). We smoothly cut off the +defining function in a 1 +n-neighborhood of the boundary to make the surface exactly catenoidal +there and then extend using these annuli on the other side of the truncation boundary all the way +to ∂B3. The result is our initial surface �Ξ−K0∪K0 +n +. The initial surface �Σ−K0∪B2∪K0 +m +is constructed +analogously, now also smoothly transitioning from the middle truncation boundary to coincide with +B2 on neighborhood of the origin. In what follows we will distill those objects and ancillary results +that are needed for the spectral convergence theorems we will prove in Section 5.4. +Decompositions. +Recalling (5.4) for the definition of the below domains, our construction in [6] +provides, in particular, smooth maps +ϕMΞ +n : MΞ +−(n5/8) → Ξ−K0∪K0 +n +, +ϕMΣ +m : MΣ +−((m + 1)5/8) → Σ−K0∪B2∪K0 +m +, +which are smooth coverings of their images. For all 0 < s ≤ √n or, respectively, 0 < s ≤ √m + 1 +we in turn define +MΞ +n (s) := ϕMΞ +n �MΞ +−(s) +� ⊂ Ξ−K0∪K0 +n +, +MΣ +m(s) := ϕMΣ +m�MΣ +−(s) +� ⊂ Σ−K0∪B2∪K0 +m +. +In practice, in addition to the upper bound required on s, we will be interested only in s greater +than a universal constant set by MΞ and MΣ: in essence we want to truncate far enough out +(in the domain) that near and beyond the truncation boundary the surface is already the graph +of a small function over the asymptotic planes. In a typical application to follow we will take s +large in absolute terms and then take n or m large with respect to s, so we will not always repeat +either restriction. When they do hold, Ξ−K0∪K0 +n +\ MΞ +n (s) consists of two connected components and +Σ−K0∪B2∪K0 +m +\ MΣ +m(s) consists of three, and we define +KΞ +n (s) := the closure of the component of Ξ−K0∪K0 +n +\ MΞ +n (s) on which z is maximized, +KΣ +m(s) := the closure of the component of Σ−K0∪B2∪K0 +m +\ MΣ +m(s) on which z is maximized, +BΣ +m(s) := the closure of the component of Σ−K0∪B2∪K0 +m +\ MΣ +m(s) that contains the origin. +Observe that each MΞ +n (s) is invariant under R{z=0}, that the interiors of MΞ +n (s), KΞ +n (s), and +R{z=0}KΞ +n (s) are pairwise disjoint, and that the last three regions cover Ξ−K0∪K0 +n +. In particular, +considering the interior of such sets, one thereby determines a candidate partition for the application +of Proposition 3.1. Similarly, MΣ +m(s) and BΣ +m(s) are invariant under R{y=z=0}; the interiors of MΣ +m(s), +BΣ +m(s), KΣ +m(s), and R{y=z=0}KΣ +m(s) are pairwise disjoint, also such four surfaces cover Σ−K0∪B2∪K0 +m +. +We agree to distinguish the choices s = √n and s = √m + 1 by omission of the parameter value: +MΞ +n := MΞ +n (√n), +KΞ +n := KΞ +n (√n), +MΣ +m := MΣ +m( +√ +m + 1), +KΣ +m := KΣ +m( +√ +m + 1), +BΣ +m := BΣ +m( +√ +m + 1), +as visualized in Figure 5. We also define the dilated truncations (cf. Figure 6) +MΞ +fb,n(s) := n +� +MΞ +n (s) ∩ W π/(2n) +−π/(2n) +� += nϕMΞ +n (MΞ +fb(s)), +MΞ +fb,n := MΞ +fb,n(√n), +MΣ +fb,m(s) := (m + 1) +� +MΣ +m(s) ∩ W π/(2(m+1)) +−π/(2(m+1)) +� += (m + 1)ϕMΣ +m(MΣ +fb(s)), +MΣ +fb,m := MΣ +fb,m( +√ +m + 1), +38 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +where the notation for wedges has been given in (4.2), and finally introduce the transition regions +ΛΞ +n(s) := MΞ +fb,n \ MΞ +fb,n(s), +ΛΣ +m(s) := MΣ +fb,m \ MΣ +fb,m(s). +Geometric estimates. +Before proceeding, we declare the following abbreviated notation for the +metrics and second fundamental forms on MΞ +fb,n and MΣ +fb,m (induced by their inclusions in (R3, gR3)): +gΞ +n := gMΞ +fb,n, +gΣ +m := gMΣ +fb,m, +AΞ +n := AMΞ +fb,n, +AΣ +m := AMΣ +fb,m. +In analogy with (5.5) we first write ψΞ +n, ψΣ +m for the unique functions on MΞ +fb,n, MΣ +fb,m such that +ψΞ = +� +n ◦ ϕMΞ +n +�∗ ψΞ +n, +ψΣ = +� +(m + 1) ◦ ϕMΣ +m +�∗ ψΣ +m +and then in turn define +ρΞ +n := +� +ψΞn + 1 +2 +��AΞn +��2 +gΞ +n(1 − ψΞn) + e−2n, +hΞ +n := (ρΞ +n)2gΞ +n, +ρΣ +m := +� +ψΣ +m + 1 +2 +��AΣm +��2 +gΣ +m(1 − ψΣ +m) + e−2m, +hΣ +m := (ρΣ +m)2gΣ +m. +(5.8) +The terms e−2n and e−2m above are included to ensure the conformal factors vanish nowhere. For +the sake of brevity, and consistently with the notation adopted in the previous subsections, we set +� +MΞ +fb,n := (MΞ +fb,n, hΞ +n), +� +MΞ +fb,n(s) := (MΞ +fb,n(s), hΞ +n) +� +MΣ +fb,m := (MΣ +fb,m, hΣ +m), +� +MΣ +fb,m(s) := (MΣ +fb,m(s), hΣ +m), +so that � +MΞ +fb,n and � +MΣ +fb,m and their truncations � +MΞ +fb,n(s) ⊂ MΞ +fb,n and MΣ +fb,m(s) ⊂ MΣ +fb,m are always +equipped with the conformal metrics hΞ +n and hΣ +m, rather than gΞ +n and gΣ +m. +Lemma 5.18 (Convergence of MΞ +fb,n(s) and MΣ +fb,m(s)). For every s > 0 there exists ms > 0 such +that for every integer m > ms +(i) the region MΣ +fb,m(s) is defined and is the diffeomorphic image under (m + 1)ϕMΣ +m of MΣ +fb(s), +(ii) (m + 1)ϕMΣ +m�MΣ +fb(s) ∩ {x = 0} +� = MΣ +fb,m(s) ∩ (m + 1)S2, +(iii) ϕMΣ +m commutes with R{z=0}, and +(iv) MΣ +m(s) = (m + 1)−1Am+1MΣ +fb,m(s) is a surface with smooth boundary. +Moreover, for every s > 0 and α ∈ ]0, 1[ +(v) +� +(m + 1) ◦ ϕMΣ +m +�∗ gΣ +m +C1,α(MΣ +fb(s),gMΣ) +−−−−−−−−−−−→ +m→∞ +gMΣ and +(vi) +� +(m + 1) ◦ ϕMΣ +m +�∗ AΣ +m +C0,α(MΣ +fb(s),gMΣ) +−−−−−−−−−−−→ +m→∞ +AMΣ. +All the above statements have analogues for Ξ−K0∪K0 +n +in place of Σ−K0∪B2∪K0 +m +, mutatis mutandis. +39 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +MΞ +n +KΞ +n +BΣm +KΣ +m +MΣ +m +Figure 5: Decomposition of Ξ−K0∪K0 +n +(left) and Σ−K0∪B2∪K0 +m +(right, cutaway view). +x +MΞ +fb,n +x +MΣ +fb,m +Figure 6: The dilated truncations MΞ +fb,n (left) and MΣ +fb,m (right). +40 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +The first four claims are immediate from the definitions, while the convergence assertions are ensured, +in the case of Σ−K0∪B2∪K0 +m +, by the following estimates from [6], the case of Ξ−K0∪K0 +n +being completely +analogous. Namely, the estimate [6, (5.20)] provides C2,α bounds for the defining function of +Σ−K0∪B2∪K0 +m +as a graph over the corresponding initial surface, so controlling the projection map ϖΣ +m +from Σ−K0∪B2∪K0 +m +to the initial surface. The same estimate [6, (5.20)] also bounds the parameter +value for the initial surface from the one-parameter family that is selected to produce the final +one. On the other hand, [6, Proposition 3.18] provides estimates on the initial surface, in terms +of the datum g as well as the value of the continuous parameter. (As an aid to extracting the +required information, we point out that the map ϖMm,ξ in [6, (3.43)] is essentially (that is: up to +some quotienting and the exact extent of the domains) the inverse of the map ϖΣ +m−1 ◦ ϕMΣ +m−1 of the +present article.) +Let us consider the other portions of our surfaces. By construction ϖΞ +n(KΞ +n ) and ϖΣ +m(KΣ +m) (subsets +of the initial surfaces) are graphs (under the Euclidean metric gR3) over subsets of KϵΞ(n) and +KϵΣ(m), and ϖΣ +m(BΣ +m) a graph over B2. Thus, by composition with a further projection, we obtain +injective maps ϖΞ +n(KΞ +n ) → KϵΞ(n), ϖΣ +m(KΣ +m) → KϵΣ(m), and BΣ +m → B2. Moreover, the image of each +of these three maps is O(2) invariant: the image of the third is a disc with radius tending to 1 as +m → ∞, the image of the second is a catenoidal annulus with upper boundary circle coinciding +with that of KϵΣ(m) and lower boundary circle tending to that of KϵΣ(m) as m → ∞; the image of +the first admits an analogous description. +In particular, by composing further with dilations of scale factor tending to 1, we obtain diffeomor- +phisms +ϕBΣ +m : B2 → BΣ +m; +similarly reparametrizing in the radial direction one also obtains diffeomorphisms +ϕKΞ +n : K0 → KΞ +n , +ϕKΣ +m : K0 → KΣ +m. +The inverses of these maps may be regarded as small perturbations (for n and m large) of nearest- +point projection onto R2 ⊂ B2 or onto the complete catenoid containing K0, as appropriate. +Somewhat more formally, by reference to [6] (specifically Proposition 3.18 and estimate (5.20) +therein), much as in the proof of Lemma 5.18, we confirm the following properties of KΞ +n , KΣ +m, and +BΣ +m. +Lemma 5.19 (Convergence of KΞ +n and KΣ +m). There exists m0 > 0 such that for each integer +m > m0 +(i) ϕKΣ +m is defined and a diffeomorphism from K0 onto KΣ +m, +(ii) ϕKΣ +m commutes with each element of Ym+1, and +(iii) ϕKΣ +m takes the upper boundary component of K0 to the upper boundary component of KΣ +m. +Moreover, for every α ∈ ]0, 1[ +(iv) (ϕKΣ +m)∗gΣ−K0∪B2∪K0 +m +��� +KΣ +m +C1,α(K0,gK0) +−−−−−−−−→ +m→∞ +gK0 and +(v) (ϕKΣ +m)∗AΣ−K0∪B2∪K0 +m +��� +KΣ +m +C0,α(K0,gK0) +−−−−−−−−→ +m→∞ +AK0. +41 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +All the above statements have analogues for Ξ−K0∪K0 +n +in place of Σ−K0∪B2∪K0 +m +, mutatis mutandis. +Lemma 5.20 (Convergence of BΣ +g ). There exists m0 > 0 such that for each integer m > m0 +(i) ϕBΣ +m is defined and a diffeomorphism from B2 onto BΣ +m and +(ii) ϕBΣ +m commutes with each element of Am+1. +Moreover, for each α ∈ ]0, 1[ +(iii) (ϕBΣ +m)∗gΣ−K0∪B2∪K0 +m +��� +BΣ +m +C1,α(B2,gB2) +−−−−−−−−→ +m→∞ +gB2 and +(iv) (ϕBΣ +m)∗AΣ−K0∪B2∪K0 +m +��� +BΣ +m +C0,α(B2,gB2) +−−−−−−−−→ +m→∞ +0. +Last we focus on the transition regions. Let us agree to write tΞ +n and tΣ +m for the distance functions on +nKϵΞ(n) and (m + 1)KϵΣ(m) from their respective lower boundary circles. By construction (assuming +s large enough in absolute terms) nϖΞ +n(n−1ΛΞ +n(s)) has two connected components, one a graph over +the catenoidal annular wedge +{s ≤ tΞ +n ≤ √n} ∩ W π/(2n) +−π/(2n) ⊂ nKϵΞ(n) +and the other the reflection of this last one through {z = 0}, while (m + 1)ϖΣ +m((m + 1)−1ΛΣ +n(s)) +has three connected components, one a graph over the planar annular wedge +{s ≤ (m + 1) − r ≤ +√ +m + 1} ∩ W π/(2(m+1)) +−π/(2(m+1)) ∩ (m + 1)B2, +another a graph over the catenoidal annular wedge +{s ≤ tΣ +m ≤ +√ +m + 1} ∩ W π/(2(m+1)) +−π/(2(m+1)) ⊂ (m + 1)KϵΣ(m), +and the third the reflection of this last one through {y = z = 0}. +Projecting onto these rotationally invariant sets and parametrizing them by arc length t in the +“radial” direction and ϑ := nθ or, respectively, ϑ := (m + 1)θ, in the angular direction (with θ +restricted to the appropriate interval containing 0), we obtain injective maps +ϕΛΞ +n(s),K : +� +s, √n +� +× +� +−π +2 , π +2 +� +→ ΛΞ +n, +ϕΛΣ +m(s),K, ϕΛΣ +m(s),B2 : +� +s, +√ +m + 1 +� +× +� +−π +2 , π +2 +� +→ ΛΣ +m +whose images are components of ΛΞ +n(s) and ΛΣ +m(s) that generate the latter regions under {z = 0} +and {y = z = 0} respectively. +Lemma 5.21 (Estimates on ΛΞ +n(s) and ΛΣ +m(s)). Let α ∈ ]0, 1[. There exists s0 > 0 such that for +each s > s0 there exists ms > 0 such that for every integer m > ms +(i) (ϕΛΣ +m(s),K)∗|AΣ +m|2 +gΣ +m(t, ϑ) = a1(t)m−2 + a2(t, ϑ)e−t/4 for some smooth functions a1, a2 having +C0,α(dt2 + dϑ2) norm bounded independently of m and s, +42 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +(ii) (ϕΛΣ +m(s),B2)∗|AΣ +m|2 +gΣ +m(t, ϑ) = a3(t, ϑ)e−t/4 for some smooth function a3 having C0,α(dt2 + dϑ2) +norm bounded independently of m and s, +(iii) (ϕΛΣ +m(s),K)∗gΣ +m = dt2 + (1 + m−1tf1(t))dϑ2 + f1 +uv(t, ϑ)e−t/4 du dv for some smooth functions +f1, f1 +uv having C1,α(dt2 + dϑ2) norm bounded independently of m and s, +(iv) (ϕΛΣ +m(s),B2)∗gΣ +m = dt2 + (1 + m−1tf2(t))dϑ2 + f2 +uv(t, ϑ)e−t/4 du dv for some smooth functions +f2, f2 +uv having C1,α(dt2 + dϑ2) norm bounded independently of m and s, +(v) ∆(ϕΛΣ +m(s),K)∗gΣ +m = ∂2 +t + m−1ct +1(t)∂t + (1 + m−1/2bϑϑ +1 (t))∂2 +ϑ + e−t/4(buv +2 (t, ϑ)∂u∂v + cu +2(t, ϑ)∂u) for +some smooth functions bϑϑ +1 , buv +2 , ct +1, cu +2 having C0,α(dt2 + dϑ2) norm bounded independently of +m and s, and +(vi) ∆(ϕΛΣ +m(s),B2)∗gΣ +m = ∂2 +t + m−1ct +3(t)∂t + (1 + m−1/2bϑϑ +3 (t))∂2 +ϑ + e−t/4(buv +4 (t, ϑ)∂u∂v + cu +4(t, ϑ)∂u) for +some smooth functions bϑϑ +3 , buv +4 , ct +3, ci +4 having C0,α(dt2 + dϑ2) norm bounded independently of +m and s. +It is understood that, in items (iii), (iv), (v), (vi) one takes u, v ∈ {t, ϑ}. +Furthermore, +(vii) lim +s→∞ lim +m→∞H 2(hΣ +m)(ΛΣ +m(s)) = 0. +The same claims hold for ΛΞ +n(s), mutatis mutandis. +Proof. Again the estimates are ultimately justified by reference to the construction [6], most +specifically (5.20) and Proposition 3.18 therein. That said, we also note how claim (v) follows +easily from (iii), as does claim (vi) from (iv); furthermore, it is clear that the justification of (ii) is +analogous to (in fact simpler than) (i), and (iv) is analogous to (iii). As a result, we briefly explain +the ideas behind the elementary computations required for the proof, in the case of ΛΣ +m(s), so with +regard to items (i) and (iii). +The projection of this region onto the blown-up initial surface (m+1)�Σ−K0∪B2∪K0 +m +is itself constructed +as a graph over (m+1)KϵΣ(m) or B2. Estimate [6, (5.20)] ensures that mϵΣ(m) is bounded uniformly +in m. The defining function of the above graph is obtained by “transferring” the defining functions +of the corresponding ends of MΣ over their asymptotic planes. These defining functions decay +exponentially in the distance along the planes. In turn ΛΣ +m(s) is a graph over this portion of the +initial surface with defining function that is also guaranteed (by [6, (5.20)]) to decay exponentially, +though a priori at a slower rate; we have chosen 1/4 somewhat arbitrarily. This accounts for all +exponential factors appearing in the estimates. +The m-dependent terms in the estimates for the metric (and Laplacian) arise simply from the choice +of (t, ϑ) coordinates on disc and catenoidal models. The m−2 term in the first item arises from +scaling the second fundamental form of the “asymptotic” catenoid to this component (while the +corresponding term for the disc vanishes). With the estimates for the second fundamental form in +place, the final item – the area estimate – follows (recalling the definitions (5.8)) from the bound +� π/2 +−π/2 +� √m+1 +s +�a1m−2 + a2e−t/4� dt dϑ ≤ C +� +m−3/2 + e−s/4� +, +and the analogous estimate concerning the disk-type component instead. +43 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +5.4 Regionwise spectral convergence +For each region S among MΞ +n , MΣ +m, KΞ +n , KΣ +n , and BΣ +m (depicted in Figure 5) we write QS +N for the +Jacobi form of S as a minimal surface in B3 with boundary, subject to the Robin condition (4.1) +where ∂S meets ∂B3 and subject to the Neumann condition elsewhere: recalling (2.17), we set +QS +N := +� +� +� +� +� +� +� +� +QΞ−K0∪K0 +n +�Nint +S +for S ⊂ Ξ−K0∪K0 +n +� +QΣ−K0∪B2∪K0 +m +�Nint +S +for S ⊂ Σ−K0∪B2∪K0 +m +(where on the right-hand side we slightly abuse notation in that in place of S we really mean +its interior). Similarly, for S either MΞ +fb,n or MΣ +fb,m we write QS +N for the Jacobi form of S as a +minimal surface in either nB3 or (m + 1)B3, subject to the Robin condition either du(η) = n−1u +or du(η) = (m + 1)−1u where ∂S meets either nS2 or (m + 1)S2, respectively, and subject to the +Neumann condition elsewhere. Keeping in mind the statement of Theorem 3.1, we stress that the +adjunction of Neumann conditions in the “interior” boundaries is motivated by our task of deriving +upper bounds on the Morse index of our examples. Recalling the notation � +MΞ +fb,n and � +MΣ +fb,n, we +remark that the bilinear forms QS +N and Q�S +N agree by definition for each S as above, but whenever +we refer to the eigenvalues, eigenfunctions, index, and nullity of the latter we shall always mean +those defined with respect to the hΞ +n or hΣ +m metric. +In the notation of (2.4) we have in particular (cf. Proposition 3.11) +Q +MΞ +fb,n +N += T +� +MΞ +fb,n, gΞ +n, qΞ +n = |AΞ +n|2 +gΞ +n, rΞ +n = n−1, +∂DMΞ +fb,n = ∅, ∂NMΞ +fb,n = ∂MΞ +fb,n \ nS2, ∂RMΞ +fb,n = ∂MΞ +fb,n \ ∂NMΞ +fb,n +� += T +� +MΞ +fb,n, hΞ +n, +� +ρΞ +n +�−2 +qΞ +n, +� +ρΞ +n +�−1 +n−1, ∅, ∂MΞ +fb,n \ nS2, ∂MΞ +fb,n \ ∂NMΞ +fb,n +� += Q +� +MΞ +fb,n +N +(5.9) +and similarly for Q +MΣ +fb,m +N += Q +� +MΣ +fb,m +N +. Observe further (cf. Lemma 3.5 and Proposition 3.11) +indPn(QMΞ +n +N ) = ind+ +{z=0} +� +Q +� +MΞ +fb,n +N +� +, +indYn(QMΞ +n +N ) = ind +� +Q +� +MΞ +fb,n +N +� +, +indAm+1(QMΣ +m +N +) = ind− +{y=z=0} +� +Q +� +MΣ +fb,m +N +� +, +indYm+1(QMΣ +m +N +) = ind +� +Q +� +MΣ +fb,m +N +� +, +and likewise for the corresponding nullities. +Lemma 5.22 (Equivariant index and nullity on KΞ +n , KΣ +m, and BΣ +m). There exist n0, m0 > 0 such +that we have the following indices and nullities for all integers n > n0 and m > m0: +S +G +indG(QS +N) +nulG(QS +N) +KΞ +n +Yn +1 +0 +KΣ +m +Ym+1 +1 +0 +BΣ +m +Am+1 +0 +0 +44 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Additionally, still assuming m > m0 we have the upper bound +indYm+1 +� +QBΣ +m +N +� ++ nulYm+1 +� +QBΣ +m +N +� +≤ 1. +Proof. We use the convergence described in Lemma 5.19 and Lemma 5.20 along with Proposition 3.8 +to compare the low eigenvalues of the regions in question with those of their limiting models, as +recorded in Lemma 5.8 and Lemma 5.10. +While we have cut the surfaces Ξ−K0∪K0 +n +and Σ−K0∪B2∪K0 +m +in such a way that the resulting regions +KΞ +n and KΣ +m converge uniformly to K0 and likewise BΣ +m to B2, thereby securing the preceding +lemma in a straightforward fashion, the cases of MΞ +n and MΣ +m are more subtle. Our approach here +(especially the proof of eigenfunction bounds in Lemma 5.25 and their application to Lemma 5.26) +draws inspiration from the analysis Kapouleas makes of the invertibilty of the Jacobi operator on +“extended standard regions” in many gluing construction; for a specific example, concerning Scherk +towers glued to catenoids, we refer the reader to the proof of [17, Lemma 7.4]. +To proceed, recalling (2.17), for each s > 0 and each integer n (sufficiently large in terms of s) we +define +Q +� +MΞ +fb,n(s) +D +:= +� +Q +� +MΞ +fb,n +N +�Dint +� +MΞ +fb,n(s) +and +Q +� +MΞ +fb,n(s) +N +:= +� +Q +� +MΞ +fb,n +N +�Nint +� +MΞ +fb,n(s) +and analogously for � +MΣ +fb,m(s) in place of � +MΞ +fb,n(s). +Lemma 5.23 (Spectral convergence for � +MΞ +fb,n(s) and � +MΣ +fb,m(s)). With the above notation, we have +λ{z=0},± +i +� +Q� +MΞ +fb +� += lim +s→∞ lim +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n(s) +D +� += lim +s→∞ lim +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n(s) +N +� +for each integer i ≥ 1 and each common choice of sign ± on both sides of each equation. The +analogous statements hold, mutatis mutandis, for � +MΣ +fb,m in place of � +MΞ +fb,n. +Proof. Fix i. By Lemma 5.18 and Proposition 3.8 for each s > 0 we have +lim +n→∞ λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +D +� += λ{z=0},+ +i +� +Q� +MΞ +fb(s) +D +� +, +lim +n→∞ λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +N +� += λ{z=0},+ +i +� +Q� +MΞ +fb(s) +N +� +. +An application of Lemma 5.17 completes the proof in this case, and the proofs of the remaining +three cases are structurally identical to this one. +45 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Lemma 5.24 (Eigenvalue upper bounds on � +MΞ +fb,n and � +MΣ +fb,m). With the above notation, we have +lim sup +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n +N +� +≤ λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim sup +m→∞ λ{y=z=0},± +i +� +Q +� +MΣ +fb,m +N +� +≤ λ{y=z=0},± +i +� +Q� +MΣ +fb +� +for each integer i ≥ 1 and each common choice of sign ± on both sides of each equation. +Proof. We give the proof for the + choice on both sides of the top equation, the proofs for the +remaining three cases being identical in structure to this one. Fix i ≥ 1. By (2.13), considering +extensions by zero of functions corresponding to the right-hand side below to obtain valid test +functions corresponding to the left, we get at once the inequality +λ{z=0},+ +i +� +Q +� +MΞ +fb,n +N +� +≤ λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +D +� +for all s > 0 and all n sufficiently large in terms of s that � +MΞ +fb,n(s) is defined. We then finish by +applying Lemma 5.23. +Lemma 5.25 (Uniform bounds on eigenvalues and eigenfunctions of Q +� +MΞ +fb,n +N +and Q +� +MΣ +fb,m +N +). For each +integer i ≥ 1 there exist Ci, ki > 0 such that for each integer k > ki and whenever λ(k) +i +is the ith +eigenvalue of Q +� +MΞ +fb,k +N +or Q +� +MΣ +fb,k +N +and v(k) +i +is any corresponding eigenfunction of unit L2-norm (under +either hΞ +k or hΣ +k as appropriate), we have the bounds +max +� +|λ(k) +i +|, ∥v(k) +i +∥H1, ∥v(k) +i +∥C0 +� +≤ Ci +(where the H1 norm is defined via either hΞ +n or hΣ +m as applicable and we emphasize that Ci does not +depend on k). +Proof. We will give the proof for � +MΞ +fb,n that for � +MΣ +fb,m being identical in structure. Fix i ≥ 1 and +let λ(n) and v(n) be as in the statement for each integer n (suppressing the fixed index i); it is our +task to show that by assuming n large enough in terms of just i we can ensure the asserted bounds +on λ(n) and v(n). In particular our assumptions include the normalization ∥v(n)∥L2(MΞ +fb,n,hΞ +n) = 1. +Lemma 5.24 provides an upper bound on λ(n), independent of n. We deduce a lower bound on λ(n) +as follows. Keeping in mind the min-max characterization (2.13) we observe that in the ratio +�u, qnu +� +L2(MΞ +fb,n,hΞ +n) + +�u|∂RMΞ +fb,n, rnu|∂RMΞ +fb,n +� +L2(∂RMΞ +fb,n,hΞ +n) +∥u∥2 +L2(MΞ +fb,n,hΞ +n) +, +(5.10) +with +rn := +� +ρΞ +n +�−1 ��� +∂RMΞ +fb,n +n−1 = (1 + e−2n)−1/2n−1, +qn := +� +ρΞ +n +�−2 ���AΞ +n +��� +2 +gΞ +n +. +46 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +we have not only a uniform upper bound on rn, but also, by inspecting (5.8) and bearing in mind +the convergence described in Lemma 5.18 as well as the boundedness (with decay) of the second +fundamental form of MΞ, +sup +n +∥qn∥C0(MΞ +fb,n) < ∞. +In addition, the convergence in Lemma 5.18 further ensures that the constants appearing in (2.3), +with (Ω, g) = (MΞ +fb,n, hΞ +n) and ∂RΩ in place of ∂Ω can be chosen uniformly in n: thus, employing +such a trace inequality and exploiting the foregoing uniform bounds we secure the promised uniform +lower bound on λ(n). +In turn, from the definitions of eigenvalues and eigenfunctions and the normalization of v(n) we have +���∇hΞ +nv(n)��� +2 +L2(MΞ +fb,n,hΞ +n) = λ(n) + +� +v(n), qnv(n)� +L2(MΞ +fb,n,hΞ +n) ++ +� +v(n)|∂RMΞ +fb,n, rnv(n)|∂RMΞ +fb,n +� +L2(∂RMΞ +fb,n,hΞ +n). +The uniform bound on ∥v(n)∥H1(MΞ +fb,n,hΞ +n) now follows, in view of the above equality, from the upper +bound on λ(n) as well as again the above uniform bounds on qn and rn. +It remains to establish the uniform C0 bound. To start, by Lemma 3.5 and standard elliptic +regularity v(n) is smooth up to the boundary: indeed, it satisfies +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∆hΞ +n + +� +ρΞ +n +�−2 ���AΞ +n +��� +2 +gΞ +n ++ λ(n)� +v(n) = 0 +in MΞ +fb,n, +hΞ +n +� +ηΞ +n, ∇hΞ +nv(n)� += (1 + e−2n)−1/2n−1v(n) +on ∂RMΞ +fb,n, +hΞ +n +� +ηΞ +n, ∇hΞ +nv(n)� += 0 +on ∂NMΞ +fb,n, +(5.11) +with ηΞ +n the outward hΞ +n unit conormal to MΞ +fb,n. As established above, we have bounds independent +of n on |λ(n)| and the qn and rn functions. By Lemma 5.18 (and the uniform geometry of MΞ) we +also have uniform control over the geometry of (MΞ +fb,n(s), hΞ +n) for all s > 0 and all n sufficiently +large in terms of s. +That being said, it is convenient for us to stipulate that throughout this proof C denotes a strictly +positive constant whose value may change from instance to instance but can always be selected +independently of s and n. +Hence, first of all standard elliptic regularity therefore ensures that for every s > 0 there is ns > 0 +so that +���v(n)��� +MΞ +fb,n(s) +��� +C0(MΞ +fb,n(s),hΞ +n) ≤ C for every integer n > ns. +(5.12) +Since we do not have uniform control on the geometry of (MΞ +fb,n = MΞ +fb,n(√n), hΞ +n), we do not obtain +a global bound independent of n in the same fashion. Instead the proof will be completed by +securing a C0 bound for v(n), independent of n, on ΛΞ +n(s) for some s > 0 to be determined. +To proceed we multiply both sides of the PDE in (5.11) by +� +ρΞ +n +�2 to get +� +∆gΞ +n + +���AΞ +n +��� +2 +gΞ +n ++ λ(n) � +ρΞ +n +�2� +v(n) = 0, +(5.13) +47 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +and we aim to bound v(n) on ΛΞ +n(s) on the basis of this equation, with unknown but controlled (as +we explain momentarily) Dirichlet data on the portion of ∂ΛΞ +n(s) contained in the interior of MΞ +fb,n +and with homogeneous Neumann data on the rest of the boundary. By the symmetries it suffices to +establish the estimate on just the component of ΛΞ +n(s) that is a graph over a subset of nK0. (For +ΛΣ +m(s) one must also consider the component which is a graph over a subset of (m + 1)B2, but this +case does not differ in substance from the one we treat now.) +Recall the map +ϕΛΞ +n(s),K : +� +s, √n +� +× +� +−π +2 , π +2 +� +→ Λn(s) +introduced above Lemma 5.21 and continue to write (t, ϑ) for the standard coordinates on its +domain. For the remainder of this proof we abbreviate ϕΛΞ +n(s),K to ϕn,s and its domain to Rn,s. +Setting w(n) := ϕ∗ +n,sv(n), we pull back (5.13) to get +∆ϕ∗n,sgΞ +nw(n) = −w(n)ϕ∗ +n,s +����AΞ +n +��� +2 +gΞ +n ++ λ(n) � +ρΞ +n +�2� +. +From the uniform bound on λ(n), the expression for the conformal factor in (5.8), and item (i) of +Lemma 5.21 we in turn obtain +∆ϕ∗n,sgΞ +nw(n) = +�cn,se−t/4 + dn,sn−2�w(n) +(5.14) +for some smooth functions cn,s, dn,s having C0,α(dt2 + dϑ2) norms uniformly bounded in n and s, +with α ∈ ]0, 1[ now fixed for the rest of the proof. (Here and below when referring to items of +Lemma 5.21 we have in mind of course the corresponding statements for ΛΞ +n(s) in place of ΛΣ +m(s).) +Noting that we have (5.14) for all sufficiently large s, it now follows from the C0 bound (5.12) and +standard interior Schauder estimates (using also item (iii) of Lemma 5.21) that +���w(n)(s, ·) +��� +C2,α(dϑ2) ≤ C for every integer n > ns+1. +(5.15) +Since v(n) satisfies the homogeneous Neumann condition along ∂MΞ +fb,n, with the aid of item (iii) of +Lemma 5.21 we have +(∂tw(n))(√n, ϑ) = en,s e−√n/4(∂ϑw(n))(√n, ϑ), +(5.16) +(∂ϑw(n))(·, ±π/2) = 0 +(5.17) +for some smooth function en,s having C1,α(dt2 + dϑ2) norm bounded independently of n and s. (For +(5.17) we simply use the fact that ϕn,s has been constructed by composing and restricting maps +which commute with the symmetries of the construction, including the reflections through planes +corresponding to ϑ = ±π/2.) +Appealing again to standard Schauder estimates, now also up to the boundary, we can conclude +from (5.14), (5.15), (5.16), and (5.17) that +∥w(n)∥C2,α(dt2+dϑ2) ≤ C +�1 + ∥w(n)∥C0 +� +(5.18) +48 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +for n and s sufficiently large in terms of the bounds assumed on the functions cn,s, dn,s, and en,s, as +well as constants, which can be chosen uniformly, that appear in local Schauder estimates on Rn,s. +If we exploit (5.18) in (5.16) we get +∥(∂tw(n))(√n, ·)∥C1,α(dϑ2) ≤ Ce−√n/4�1 + ∥w(n)∥C0 +�, +(5.19) +once again for n and s assumed large enough in terms of absolute constants. +We next decompose w(n) into +w(n) +0 +:= 1 +π +� π/2 +−π/2 +w(n)(·, ϑ) dϑ, +w(n) +⊥ +:= w(n) − w(n) +0 . +From (5.14), (5.18), and item (v) of Lemma 5.21 we obtain +∂2 +t w(n) +0 += a0 +n,se−t/4 + b0 +n,sn−2 + c0 +n,sn−1∂tw(n) +0 , +with +∥a0 +n,s∥C0 + ∥b0 +n,s∥C0 +1 + ∥w(n)∥C0 ++ ∥c0 +n,s∥C0 ≤ C +(5.20) +and +∥∆dt2+dϑ2w(n) +⊥ ∥C0 ≤ C +� +e−s/4 + n−1/2��1 + ∥w(n)∥C0 +�. +(5.21) +For (5.20) we have in particular integrated (5.14) in ϑ, making use of the ϑ-invariance (see item +(v) of Lemma 5.21) of the coefficients of the n−1∂t and n−1/2∂2 +ϑ terms and observing that the +n−1/2∂2 +ϑ term integrates to zero because of (5.17); for (5.21) we have made use of the fact that +∥∆dt2+dϑ2w(n) +⊥ ∥C0 ≤ 2∥∆dt2+dϑ2w(n)∥C0 and then appealed to (5.14). +To complete the analysis we will need some basic estimates for ∆dt2+dϑ2 = ∂2 +t + ∂2 +ϑ on Rn,s. For +any bounded (real-valued) function f on Rn,s and for each non-negative integer κ let us define on +[s, √n] the Fourier coefficients fκ by +fκ(t) := +� +� +� +1 +π +� π/2 +−π/2 f(t, ϑ) dϑ +for κ = 0 +2 +π +� π/2 +−π/2 f(t, ϑ) cos κ(ϑ − π/2) dϑ +for κ > 0. +Then the Fourier coefficients of any u ∈ C2(Rn,s, dt2 + dϑ2) satisfying (∂ϑu) = 0 at ϑ = ±π/2 admit +the representations +u0(t) = u0(s) + (∂tu0)(√n) · (t − s) + +� t +s +� τ +√n +∂2 +t u0(σ) dσ dτ, += u0(s) + (∂tu0)(√n) · (t − s) + +� t +s +� τ +√n +(∆dt2+dϑ2u)0(σ) dσ dτ, +(5.22) +uκ̸=0(t) = +uκ(s) +cosh κ(√n − s) cosh κ(t − √n) + +(∂tuκ)(√n) +κ cosh κ(√n − s) sinh κ(t − s) +− cosh κ(t − √n) +κ cosh κ(√n − s) +� t +s +(∆dt2+dϑ2u)κ(τ) sinh κ(τ − s) dτ +(5.23) +− +sinh κ(t − s) +κ cosh κ(√n − s) +� √n +t +(∆dt2+dϑ2u)κ(τ) cosh κ(τ − √n) dτ. +49 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +In particular (5.23) implies, for any κ ≥ 1 the inequality +|uκ(t)| ≤ |uκ(s)| + 1 +κ|(∂tuκ)(√n)| + 1 +κ2 ∥(∆dt2+dϑ2u)κ∥C0. +(5.24) +Since u is C2 the Fourier series �∞ +κ=0 uκ(t) cos κ(ϑ − π/2) converges (at least) pointwise to u(t, ϑ); +furthermore (again appealing to the C2 assumption in order to control the first two terms of (5.24)) +we obtain the implication +� π/2 +−π/2 +u(·, ϑ) dϑ = 0 +⇓ +∥u∥C0 ≤ C +� +∥u(s, ·)∥C2(dϑ2) + ∥(∂tu)(√n, ·)∥C1(dϑ2) + ∥∆dt2+dϑ2u∥C0 +� +. +(5.25) +This last estimate in conjunction with (5.21), (5.15), and (5.19) yields +∥w(n) +⊥ ∥C0 ≤ C + C(e−√n/4 + e−s/4 + n−1/2)∥w(n)∥C0. +(5.26) +On the other hand, differentiating (5.22) with respect to t and applying (5.20) and (5.19) we find +∥∂tw(n) +0 ∥C0 ≤ C +�1 + ∥w(n)∥C0 +�� +e−√n/4 + e−s/4 + n−3/2� ++ Cn−1/2∥∂tw(n) +0 ∥C0 +and therefore, by absorption, +∥∂tw(n) +0 ∥C0 ≤ C +�1 + ∥w(n)∥C0 +�� +e−s/4 + n−3/2� +(5.27) +for n sufficiently large in terms of s and the constants appearing in the above estimate. Feeding +(5.27) into (5.20) and applying the result, along with (5.15) and (5.19), in (5.22), we get +∥w(n) +0 ∥C0 ≤ C + C(√ne−√n/4 + e−s/4 + n−1)∥w(n)∥C0. +(5.28) +Finally, since ∥w(n)∥C0 ≤ ∥w(n) +0 ∥C0 + ∥w(n) +⊥ ∥C0, estimates (5.28) and (5.26) jointly imply the desired +bound on the C0 norm of w(n) provided we first choose s and then, in turn, n sufficiently large, in +terms of the absolute constants appearing in the two estimates, to be able to absorb the ∥w(n)∥C0 +terms appearing on their right-hand sides. This ends the proof. +Lemma 5.26 (Eigenvalue lower bounds on � +MΞ +fb,n and � +MΣ +fb,m). For each integer i ≥ 1 +lim inf +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n +N +� +≥ λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim inf +m→∞ λ{y=z=0},± +i +� +Q +� +MΣ +fb,m +N +� +≥ λ{y=z=0},± +i +� +Q� +MΣ +fb +� +for each common choice of sign ± on both sides of each equation. +50 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +Proof. We give the proof for the + choice on both sides of the top equation, the argument for +the remaining three cases being identical in structure to this one. Fix i ≥ 1, and for each n let +{v(n) +j +}i +j=1 be an L2(MΞ +fb,n, hΞ +n) orthonormal set such that each v(n) +j +is a jth ({z = 0}, +)-invariant +eigenfunction of Q +� +MΞ +fb,n +N +. Fix C > 0, as afforded by Lemma 5.25, such that +sup +n +sup +1≤j≤i +� +∥v(n) +j +∥C0 + λ{z=0},+ +j +� +Q � +MΞ +fb,n +�� +≤ C. +Given any ϵ > 0 (fixed from now on) and taking s > 0 and correspondingly ns > 0 large enough, as +afforded by Lemma 5.21 and Lemma 5.23, we have +H 2(hΞ +n)(Λn(s)) < ϵ, +λ{z=0},+ +i +� +Q� +MΞ +fb +� +< λ{z=0},+ +i +� +Q +� +MΞ +fb,n(s) +N +� ++ ϵ. +(5.29) +Now, for n > ns and 1 ≤ j ≤ i we consider the restrictions +v(n,s) +j +:= v(n) +j +|MΞ +fb,n(s) +and estimate (using the notation δjk for the Kronecker delta) +���δjk − +� +v(n,s) +j +, v(n,s) +k +� +L2(MΞ +fb,n(s),hΞ +n) +��� ≤ C2ϵ, +���∇hΞ +nv(n,s) +j +��� +L2(MΞ +fb,n(s),hΞ +n) ≤ +���∇hΞ +nv(n) +j +��� +L2(MΞ +fb,n,hΞ +n), +� +v(n,s) +j +, +� +ρΞ +n +�−2���AΞ +n +��� +2 +gΞ +n +v(n,s) +j +� +L2(MΞ +fb,n(s),hΞ +n) ≥ +� +v(n) +j +, +� +ρΞ +n +�−2���AΞ +n +��� +2 +gΞ +n +v(n) +j +� +L2(MΞ +fb,n,hΞ +n) − 2C2ϵ, +where for the last inequality we have used the fact that on Λn(s) the potential function appearing +here is bounded above by 2, as is obvious from inspection of (5.8). +We conclude that for all n > ns the set {v(n,s) +j +}i +j=1 is linearly independent, and for all 1 ≤ j ≤ i +Q +� +MΞ +fb,n(s) +N +� +v(n,s) +j +, v(n,s) +j +� +��v(n,s) +j +��2 +L2(MΞ +fb,n(s),hΞ +n) +≤ +λ{z=0},+ +j +� +Q +� +MΞ +fb,n +N +� ++ 2C2ϵ +1 − C2ϵ +and so by virtue of the min-max characterization (2.13) of the eigenvalues +λ{z=0},+ +j +� +Q +� +MΞ +fb,n(s) +N +� +≤ +λ{z=0},+ +j +� +Q +� +MΞ +fb,n +N +� ++ 2C2ϵ +1 − C2ϵ +for all n > ns and 1 ≤ j ≤ i. Thus, using the second inequality in (5.29), we get in particular +λ{z=0},+ +i +� +Q� +MΞ +fb +� +≤ +λ{z=0},+ +i +� +Q +� +MΞ +fb,n +N +� ++ 2C2ϵ +1 − C2ϵ ++ ϵ +for all n > ns. The claim now follows, since this inequality holds for all ϵ > 0, with C independent +of ϵ and n. +51 + +5 Effective index estimates for two sequences of examples +A. Carlotto, M. B. Schulz, D. Wiygul +By combining Lemma 5.24 with Lemma 5.26 we immediately derive the following conclusion. +Corollary 5.27 (Eigenvalues on � +MΞ +fb,n and � +MΣ +fb,m). For each integer i ≥ 1 +lim +n→∞ λ{z=0},± +i +� +Q +� +MΞ +fb,n +N +� += λ{z=0},± +i +� +Q� +MΞ +fb +� +, +lim +m→∞ λ{y=z=0},± +i +� +Q +� +MΣ +fb,m +N +� += λ{y=z=0},± +i +� +Q� +MΣ +fb +� +, +for each common choice of sign ± on both sides of each equation. +Corollary 5.28 (Equivariant index and nullity on MΞ +n and MΣ +m). There exist n0, m0 > 0 such that +we have the following indices and nullities for all integers n > n0 and m > m0. +S +G +indG(QS +N) +nulG(QS +N) +MΞ +n +Pn +1 +0 +MΣ +m +Am+1 +1 +0 +Additionally, still assuming m > m0 we have the upper bound +indYm+1 +� +QMΣ +m +N +� ++ nulYm+1 +� +QMΣ +m +N +� +≤ 3. +Proof. All claims follow from the conjunction of Lemma 3.5 (to reduce to the appropriately even +and odd indices and nullities on n−1MΞ +fb,n and (m + 1)−1MΣ +fb,m with Neumann boundary data), +Proposition 3.11 (to dispense with the above scale factors n, m + 1 and, more substantially, to +pass from the natural metric to hΞ +n or hΣ +m), Lemma 5.27 (to reduce to the appropriate indices and +nullities of � +MΞ +fb and � +MΣ +fb ), and finally Lemma 5.16 (which provides these last quantities). +5.5 Proofs of Theorem 1.2 and 1.1 +The following statement collects, from the broader analysis conducted in the previous section, those +conclusions we shall need to prove the two main results stated in the introduction. +Corollary 5.29 (Equivariant index and nullity upper bounds for Σ−K0∪B2∪K0 +m +and Ξ−K0∪K0 +n +). There +exists m0, n0 > 0 such that for all integers m > m0 and n > n0 we have the bounds +indAm+1(Σ−K0∪B2∪K0 +m +) + nulAm+1(Σ−K0∪B2∪K0 +m +) ≤ 2, +indYm+1(Σ−K0∪B2∪K0 +m +) + nulYm+1(Σ−K0∪B2∪K0 +m +) ≤ 5, +indPn(Ξ−K0∪K0 +n +) ++ nulPn(Ξ−K0∪K0 +n +) +≤ 2. +Proof. We apply item (ii) of Proposition 3.1, for the partition “into building blocks” defined in +Section 5.3 (cf. Figure 5), in conjunction with Lemma 5.22 and Corollary 5.28 for the ancillary +estimates for the index and nullity of the various blocks. +52 + +References +A. Carlotto, M. B. Schulz, D. Wiygul +So, we are in position to fully determine the (maximally) equivariant index and nullity for the two +families of free boundary minimal surfaces we constructed in [6]. +Proof of Theorem 1.2. We combine the upper bounds of the preceding corollary with the lower +bounds from our earlier paper [6], specifically with the content of Proposition 7.1 (cf. Remark 7.5) +therein for what pertains the index. At that stage, the fact that both nullities are zero then follows +from the first and third inequality in Corollary 5.29. +Finally, we can obtain the absolute estimates on the Morse index of the same families. +Proof of Theorem 1.1. The lower bounds have already been established: specifically, for Σ−K0∪B2∪K0 +m +this is just part of Proposition 5.4, while for Ξ−K0∪K0 +n +it follows from just combining Proposition 5.4 +with Proposition 5.5. For the upper bound we can apply the Montiel–Ros argument making use +of the equivariant upper bounds above, as we are about to explain. In the case of Ξ−K0∪K0 +n +, the +Pn-equivariant upper bound on the Morse index (and nullity) is equivalent to an upper bound on +the index an nullity on each domain Ωn +i = Ξ−K0∪K0 +n +∩ Wi where W1, . . . , W4n are the open domains +defined, in B3, by the horizontal plane {z = 0} together with the 2n vertical planes passing through +the origin and having equations θ = π/(2n)+iπ/n, i = 0, 1, . . . , 2n−1 (in the cylindrical coordinates +defined at the beginning of Section 4), subject to Neumann conditions in the interior boundary +as prescribed by Lemma 3.5. Thus the conclusion comes straight by appealing to Corollary 3.2. +Similarly, for Σ−K0∪B2∪K0 +m +we can interpret the second inequality in the statement of Corollary 5.29 +as a statement on the index and nullity of the portions of surfaces that are contained in any of the +2(m + 1) sets obtained by intersecting the horizontal plane {z = 0} with the m + 1 vertical planes +passing through the origin and having equations θ = π/(2(m + 1)) + 2iπ/(m + 1), i = 0, 1, . . . , m, +again subject to Neumann conditions. This completes the proof. +References +[1] L. Ambrozio, A. Carlotto, and B. Sharp, Comparing the Morse index and the first Betti number of minimal +hypersurfaces, J. Differential Geom. 108 (2018), no. 3, 379–410. +[2] +, Index estimates for free boundary minimal hypersurfaces, Math. Ann. 370 (2018), no. 3-4, 1063–1078. +[3] N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities +of second order, J. Math. 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Schulz +University of Münster, Mathematisches Institut, Einsteinstrasse 62, 48149 Münster, Germany +E-mail address: mario.schulz@uni-muenster.de +David Wiygul +ETH D-Math, Rämistrasse 101, 8092 Zürich, Switzerland +E-mail address: david.wiygul@math.ethz.ch +55 +