diff --git "a/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/load_file.txt" "b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/1NE1T4oBgHgl3EQfRgP7/content/tmp_files/load_file.txt" @@ -0,0 +1,1794 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf,len=1793 +page_content='Spectral estimates for free boundary minimal surfaces via Montiel–Ros partitioning methods Alessandro Carlotto, Mario B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' the genus and the number of boundary components) of the surfaces in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The reader is referred to [12, 13, 27] for recent overviews of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' [28–30] and references therein) has acquired great importance within geometric analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Partly motivated by the corresponding conjectures concerning closed minimal hypersurfaces in manifolds of positive Ricci curvature (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' [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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Specifically, it was shown in [2] that the following estimate holds: index(Σ) ≥ 1 3(2g + b − 1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='03055v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='DG] 8 Jan 2023 1 Introduction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul conceivable to have a superlinear growth rate of the index with respect to g and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1, and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2) for precise statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' With this approach, lower bounds are sometimes relatively cheap to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We present one simple yet paradigmatic such result in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2, which concerns free boundary minimal surfaces with pyramidal or prismatic symmetry in B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Said otherwise, one needs to know how (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' by which method) the surface under study has been obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' As we described at length in Section 7 therein, with data (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' This is a rather unexpected behaviour (by comparison 2 1 Introduction A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In particular, we establish the following statement: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1 (Index estimates for Σ−K0∪B2∪K0 m and Ξ−K0∪K0 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The next statement thus pertains to such equivariant bounds, for which we do obtain equality, thus settling part of Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='7 (iv) and Conjecture 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='9 (iv) of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2 (Equivariant index and nullity of Σ−K0∪B2∪K0 m and Ξ−K0∪K0 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Following the general methodology of [17],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' one first considers a singular configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' that is a formal union of minimal surfaces in B3 (not necessarily free boundary),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' then its regularization – which needs the use of (wrapped) periodic minimal surfaces in R3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' to desingularize near the divisors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' and controlled interpolation processes between the building blocks in play – and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' thirdly and finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' the perturbation of such configurations to exact minimality (at least for some values of the parameters),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' while also ensuring proper embeddedness and accommodating the free boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' spherical triangles, wedges or lunes) subdomains of round S2, 3 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul with mixed boundary conditions, and possibly subject to additional symmetry requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' As a result, we need to deal with delicate scale-picking arguments, an ad hoc study of the geometry of such regions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='21) and – most importantly – prove the corresponding uniform bounds for eigenvalues and eigenfunctions (collected in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='25), which allow to rule out pathologic concentration phenomena, thereby leading to the desired conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 947923).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The research of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Part of this article was finalized while A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' was visiting the ETH-FIM, whose support and excellent working conditions are gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2 Notation and standing assumptions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Throughout this article we will in fact assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We are going to study the spectrum of a given Schrödinger operator on Ω subject to boundary conditions and, sometimes, symmetry constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Such symmetry constraints will be encoded in terms of equivariance with respect to a certain group action, which we shall specify at due place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The Schrödinger operator ∆g + q is determined by the data of a given smooth Riemannian metric g on Ω and a given smooth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' C∞) function q: Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' To avoid ambiguities, we remark here that a function (or tensor field) on 4 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul Ω smooth if it is the restriction of a smooth tensor field on M or – equivalently – on a relatively open set containing Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The boundary conditions are specified by another smooth function r: Ω → R and a decomposition ∂Ω = ∂DΩ ∪ ∂NΩ ∪ ∂RΩ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1) where the sets on the right-hand side are the closures of pairwise disjoint open subsets ∂DΩ, ∂NΩ, and ∂RΩ of ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2) where ηΩ g is the almost-everywhere defined outward unit normal induced by g on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' ∇gu is uniquely defined whenever it exists, modulo vector fields vanishing almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (This can be deduced, for example, by inspecting the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='6 in [9]: 5 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In practice we will consider traces on just ∂DΩ and ∂RΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='6) where, for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='6), one can take C(Ω, g, q, r) = ∥q∥C0(Ω) + C(Ω, g)∥r∥C0(∂RΩ), thanks to the trace inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul for all f1, f2 ∈ L2(Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' That being clarified, to improve readability we will from now on refrain from explicitly indicating the inclusion map ι in our equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='7) and we call any such u an eigenfunction of T with eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (We caution that the notions of eigenfunctions and eigenvalues depend not only on T but also on the underlying metric g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' for the sake of convenience we choose to suppress the latter dependence from our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (To avoid ambiguities, we remark that the phrase Hilbertian basis refers to a countable, complete orthonormal system for the Hilbert space in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=') 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Of course E∼t(T) is a subspace of H1 ∂DΩ(Ω, g) – in particular – whenever the former has finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Taking ∼ to be equality clearly reproduces the originally defined space E=t(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' For future use observe that the above spectral theorem for T implies (E∼t(T))⊥L2(Ω,g) = E̸∼t(T), E, ≥, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='9) 7 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='9) ∼ is any relation on R and ̸∼ its negation (so that {s ̸∼ t} = R \\ {s ∼ t} for any t ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Index and nullity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Such invariants will be of primary interest in our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='4 Group actions Let G be a finite group of smooth diffeomorphisms of M, each restricting to a surjective isometry of (Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We say that a function u is G-invariant if it is invariant under this action: equivalently u ◦ φ = u for all φ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Obviously the above standard action (φ, u) �→ u ◦ φ−1 is recovered by taking the trivial homomorphism σ ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' a group homomorphism �σ: G → O �L2(Ω, g) � φ �→ σ(φ)φ−1∗ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='10) whose target are the global isometries of L2(Ω, g);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' we note that the same conclusions hold true with H1(Ω, g) in place of L2(Ω, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The corresponding subspaces of (G, σ)-invariant functions, in L2(Ω, g) 8 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul or H1(Ω, g), are readily checked to be closed, and thus Hilbert spaces themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' That said, we define the orthogonal projection πG,σ : L2(Ω, g) → L2(Ω, g) u �→ 1 |G| � φ∈G �σ(φ)u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='11) Here |G| is the order of G, which – we recall – is assumed throughout to be finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The image of L2(Ω, g) under πG,σ thus consists of (G, σ)-invariant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='11) with the average over G with respect to its Haar measure (which reduces to the former for finite G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Henceforth we make the additional assumptions that G globally (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' as sets) preserves each of ∂DΩ, ∂NΩ, and ∂RΩ, and that q and r are both G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='12) is a subspace of E∼t(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' equivalently one can work with spaces of (G, σ)-invariant functions and derive the analogous conclusions as in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3 directly in that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' This type of phenomenon happens, for instance, when every point of the manifold M is a fixed point of a σ-odd isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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) ≤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' will in fact just be a finite sequence, consisting say of I(G, σ) elements, counted with multiplicity as usual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' we shall formally convene that λG,σ i (T) = +∞ for i > I(G, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='13) 9 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Obviously we can recover E∼t(T), λi (T), and the standard index and nullity by taking G to be the trivial group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We digress briefly to highlight two important special cases, which warrant additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='3 (Actions of order-2 groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' When |G| = 2, there are precisely two homomorphisms G → O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Considering such homomorphisms, and the corresponding (G, σ)-invariant functions, we may define G-even or G-odd functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Hence, we may call ind+ G and ind− G the G-even and G-odd index, and likewise for the nullity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Clearly, we always have � � � ind(T) = ind+ G(T) + ind− G(T), nul(T) = nul+ G(T) + nul− G(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='14) Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='4 (Actions of self-congruences of two-sided hypersurfaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We shall further assume that the action of G on Ω is faithful, meaning that only the identity element fixes Ω pointly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' this assumption is always satisfied in our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Similarly, in this context, we set indG(T) := indsgnν G (T) and nulG T := nulsgnν G (T), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='15) which we may refer to as simply the G-equivariant index and G-equivariant nullity of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' We now return to the more general assumptions on G preceding this paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 10 2 Notation and standing assumptions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='5 Subdomains Suppose that Ω1 ⊂ Ω is another Lipschitz domain of M (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='17) defined, respectively, on the Sobolev spaces H1 ∂Dint D Ω1(Ω1, g) and H1 ∂Nint D Ω1(Ω1, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' since G preserves Ω and respects the decomposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1), it follows that it also respects the decompositions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Somewhat abusively, we shall write �σ and πG,σ not only for the maps (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='10) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='11) but also for their counterparts with Ω replaced by Ω1, which are well-defined under our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' The spaces E∼t G,σ(T Dint Ω1 ) and E∼t G,σ(T Nint Ω1 ) as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='12), are then also well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' ∂RΩ ∂DΩ ∂DΩ ∂NΩ ∂NΩ Ω1 ∂extΩ1 ∩ ∂NΩ ∂extΩ1 ∩ ∂DΩ ∂extΩ1 ∩ ∂RΩ ∂intΩ1 Figure 1: Example of a Lipschitz domain Ω with subdomain Ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 11 3 Fundamental tools A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Carlotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Schulz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Wiygul 3 Fundamental tools 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='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, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' , Ωn ⊂ Ω which are pairwise disjoint, each of which satisfies the same assumptions as Ω1 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='5, and whose closures cover Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' In particular, we assume that each element of the group G maps each subdomain Ωi onto itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' 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.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content='1 (Montiel–Ros bounds on the number of eigenvalues below a threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE1T4oBgHgl3EQfRgP7/content/2301.03055v1.pdf'} +page_content=' With assumptions as in the preceding paragraph and notation as in Section 2, the following inequalities hold for any t ∈ R (i) dim E