arXiv subjects

Jean Lagacé

Publications and source records attributed to Jean Lagacé.

15 recordsLinked to original sources

arXiv2026

Metric-uniform spectral inequality for the Laplacian on manifolds with bounded sectional curvature

Given a Riemannian manifold $M$ endowed with a smooth metric $g$ satisfying upper and lower sectional curvature bounds, we show an equivalence property between the $\mathrm{L}^2$ norm on $M$ and the $\mathrm{L}^2$ norm on subsets $ω$ satisfying a thickness condition, for functions in the range of a spectral projector. The thickness condition is known to be optimal in this setting. The constant appearing in the equivalence of norms property depends only on the dimension of the manifold, curvature bounds, and frequency threshold of the spectral cutoff, but, crucially, not on the injectivity radius.

math.AP↗

arXiv2024

Canonical foliation of bubblesheets

We introduce a new curvature condition for high-codimension submanifolds of a Riemannian ambient space, called quasi-parallel mean curvature (QPMC). The class of submanifolds with QPMC includes all CMC hypersurfaces and submanifolds with parallel mean curvature. We use our notion of QPMC to prove that certain kinds of high-curvature regions which appear in geometric flows, called bubblesheets, can be placed in a suitable normal form. This follows from a more general result asserting that the manifold $\mathbb{R}^k \times \mathbb{S}^{n-k}$, equipped with any metric which is sufficiently close to the standard one, admits a canonical foliation by embedded $(n-k)$-spheres with QPMC.

math.DG↗

arXiv2024

Sharp density discrepancy for cut and project sets: An approach via lattice point counting

Cut and project sets are obtained by taking an irrational slice of a lattice and projecting it to a lower dimensional subspace, and are fully characterised by the shape of the slice (window) and the choice of the lattice. In this context we seek to quantify fluctuations from the asymptotics for point counts. We obtain uniform upper bounds on the discrepancy depending on the diophantine properties of the lattice as well as universal lower bounds on the average of the discrepancy. In an appendix, Michael Björklund and Tobias Hartnick obtain lower bounds on the $L^2$-norm of the discrepancy also depending on the diophantine class; these lower bounds match our uniform upper bounds and both are therefore sharp. Using the sufficient criteria of Burago--Kleiner and Aliste-Prieto--Coronel--Gambaudo we find an explicit full-measure class of cut and project sets that are biLipschitz equivalent to lattices; the lower bounds on the variance indicate that this is the largest class of cut and project sets for which those sufficient criteria can apply.

math.NT↗

arXiv2024

Spectral decomposition and Siegel-Veech transforms for strata: The case of marked tori

Generalizing the well-known construction of Eisenstein series on the modular curves, Siegel-Veech transforms provide a natural construction of square-integrable functions on strata of differentials on Riemannian surfaces. This space carries actions of the foliated Laplacian derived from the SL(2,R)-action as well as various differential operators related to relative period translations. In the paper we give spectral decompositions for the stratum of tori with two marked points. This is a homogeneous space for a special affine group, which is not reductive and thus does not fall into well-studied cases of the Langlands program, but still allows to employ techniques from representation theory and global analysis. Even for this simple stratum exhibiting all Siegel-Veech transforms requires novel configurations of saddle connections. We also show that the contiunuous spectrum of the foliated Laplacian is much larger than the space of Siegel-Veech transforms, as opposed to the case of the modular curve. This defect can be remedied by using instead a compound Laplacian involving relative period translations.

math.NT↗

arXiv2022

Flexibility of Steklov eigenvalues via boundary homogenisation

Recently, D. Bucur and M. Nahon used boundary homogenisation to show the remarkable flexibility of Steklov eigenvalues of planar domains. In the present paper we extend their result to higher dimensions and to arbitrary manifolds with boundary, even though in those cases the boundary does not generally exhibit any periodic structure. Our arguments use framework of variational eigenvalues and provides a different proof of the original results. Furthermore, we present an application of this flexibility to the optimisation of Steklov eigenvalues under perimeter constraint. It is proved that the best upper bound for normalised Steklov eigenvalues of surfaces of genus zero and any fixed number of boundary components can always be saturated by planar domains. This is the case even though any actual maximiser (except for simply connected surfaces) is always far from being planar themselves. In particular, it yields sharp upper bound for the first Steklov eigenvalue of doubly connected planar domains.

math.SP↗

arXiv2022

Weyl's law for the Steklov problem on surfaces with rough boundary

The validity of Weyl's law for the Steklov problem on domains with Lipschitz boundaries is a well-known open question in spectral geometry. We answer this question in two dimensions and show that Weyl's law holds for an even larger class of surfaces with rough boundaries. This class includes domains with interior cusps as well as 'slow' exterior cusps. Moreover, the condition on the speed of exterior cusps cannot be improved, which makes our result in a sense optimal. The proof is based on the methods of Suslina and Agranovich combined with some observations about the boundary behaviour of conformal mappings.

math.SP↗

arXiv2021

The almost periodic gauge transform -- An abstract scheme with applications to Dirac Operators

One of the main tools used to understand both qualitative and quantitative spectral behaviour of periodic and almost periodic Schrödinger operators is the method of gauge transform. In this paper, we extend this method to an abstract setting, thus allowing for greater flexibility in its applications that include, among others, matrix-valued operators. In particular, we obtain asymptotic expansions for the density of states of certain almost periodic systems of elliptic operators, including systems of Dirac type. We also prove that a range of periodic systems including the two-dimensional Dirac operators satisfy the Bethe--Sommerfeld property, that the spectrum contains a semi-axis -- or indeed two semi-axes in the case of operators that are not semi-bounded.

math-ph↗

arXiv2021

Spectral invariants of Dirichlet-to-Neumann operators on surfaces

We obtain a complete asymptotic expansion for the eigenvalues of the Dirichlet-to-Neumann maps associated with Schrödinger operators on compact Riemannian surfaces with boundary. For the zero potential, we recover the well-known spectral asymptotics for the Steklov problem. For nonzero potentials, we obtain new geometric invariants determined by the spectrum of what we call the parametric Steklov problem. In particular, for constant potentials parametric Steklov problem, the total geodesic curvature on each connected component of the boundary is a spectral invariant. Under the constant curvature assumption, this allows us to obtain some interior information from the spectrum of these boundary operators.

math.SP↗

arXiv2020

Continuity of eigenvalues and shape optimisation for Laplace and Steklov problems

We associate a sequence of variational eigenvalues to any Radon measure on a compact Riemannian manifold. For particular choices of measures, we recover the Laplace, Steklov and other classical eigenvalue problems. In the first part of the paper we study the properties variational eigenvalues and establish a general continuity result, which shows for a sequence of measures converging in the dual of an appropriate Sobolev space, that the associated eigenvalues converge as well. The second part of the paper is devoted to various applications to shape optimization. The main theme is studying sharp isoperimetric inequalities for Steklov eigenvalues without any assumption on the number of connected components of the boundary. In particular, we solve the isoperimetric problem for each Steklov eigenvalue of planar domains: the best upper bound for the $k$-th perimeter-normalised Steklov eigenvalue is $8πk$, which is the best upper bound for the $k$-th area-normalised eigenvalue of the Laplacian on the sphere. The proof involves realising a weighted Neumann problem as a limit of Steklov problems on perforated domains. For $k = 1$, the number of connected boundary components of a maximizing sequence must tend to infinity, and we provide a quantitative lower bound on the number of connected components. A surprising consequence of our analysis is that any maximizing sequence of planar domains with fixed perimeter must collapse to a point.

math.SP↗

arXiv2020

Large Steklov eigenvalues via homogenisation on manifolds

Using methods in the spirit of deterministic homogenisation theory we obtain convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian manifold to weighted Laplace eigenvalues of that manifold. The domains are obtained by removing small geodesic balls that are asymptotically densely uniformly distributed as their radius tends to zero. We use this relationship to construct manifolds that have large Steklov eigenvalues. In dimension two, and with constant weight equal to 1, we prove that Kokarev's upper bound of $8π$ for the first nonzero normalised Steklov eigenvalue on orientable surfaces of genus 0 is saturated. For other topological types and eigenvalue indices, we also obtain lower bounds on the best upper bound for the eigenvalue in terms of Laplace maximisers. For the first two eigenvalues, these lower bounds become equalities. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than $2π$. This was previously thought to be impossible. We provide numerical evidence that some of the already known examples of free boundary minimal surfaces have these properties and also exhibit simulations of new free boundary minimal surfaces of genus 0 in the unit ball with even larger area. The first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they verify a general conjecture by Fraser and Li. In dimension three and larger, we prove that the isoperimetric inequality of Colbois--El Soufi--Girouard is sharp and implies an upper bound for weighted Laplace eigenvalues. We also show that in any manifold with a fixed metric, one can construct by varying the weight a domain with connected boundary whose first nonzero normalised Steklov eigenvalue is arbitrarily large.

math.SP↗

arXiv2020

From Steklov to Neumann via homogenisation

We study a new link between the Steklov and Neumann eigenvalues of domains in Euclidean space. This is obtained through an homogenisation limit of the Steklov problem on a periodically perforated domain, converging to a family of eigenvalue problems with dynamical boundary conditions. For this problem, the spectral parameter appears both in the interior of the domain and on its boundary. This intermediary problem interpolates between Steklov and Neumann eigenvalues of the domain. As a corollary, we recover some isoperimetric type bounds for Neumann eigenvalues from known isoperimetric bounds for Steklov eigenvalues. The interpolation also leads to the construction of planar domains with first perimeter-normalized Stekov eigenvalue that is larger than any previously known example. The proofs are based on a modification of the energy method. It requires quantitative estimates for norms of harmonic functions. An intermediate step in the proof provides a homogenisation result for a transmission problem.

math.AP↗

arXiv2019

Optimal unions of scaled copies of domains and Pólya's conjecture

Given a bounded Euclidean domain $Ω$, we consider the sequence of optimisers of the $k^{\rm th}$ Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of $Ω$ with fixed total volume. We show that this sequence encodes information yielding conditions for $Ω$ to satisfy Pólya's conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether Pólya's conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Pólya's original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa's bound in terms of packing density.

math.SP↗

arXiv2018

Eigenvalue optimisation on flat tori and lattice points in anisotropically expanding domains

This paper is concerned with the maximisation of the k'th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension d as k goes to infinity. We show that in any dimension maximisers exist for any given k, but that any sequence of maximisers degenerates as k goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the k'th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

math.SP↗

arXiv2017

The Steklov spectrum of cuboids

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids.

math.SP↗

arXiv2016

A generalised Gauss circle problem and integrated density of states

Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. The first term is the volume of the ball; we study the size of the remainder term. While the classical problem is equivalent to counting eigenvalues of the Laplace operator on the torus, our variation corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. The asymptotics we obtain are then used to compute the density of states of the magnetic Schroedinger operator.

math.SP↗