arXiv subjects

Isabel Salavessa

Publications and source records attributed to Isabel Salavessa.

8 recordsLinked to original sources

arXiv2026

Spherical n-lunes: billiards and eigenvalues

We characterise the periodic orbits of geodesic billiards on spherical lunes on $\mathbb{S}^{n}$. In the case of angle openings of the form $π/p$ for positive integer $p$ we fully determine their Dirichlet and Neumann spectra. We then show that lunes with an angle opening smaller than $π$ which is not a rational multiple of $π$, or those with an angle opening of the form $π/p$ for $p$ larger than one satisfy Pólya's conjecture eventually, independently of whether the corresponding geodesic billiards satisfy the nonperiodicity condition or not. For lunes with an angle opening $π/p$ we further provide a two-term asymptotic formula for the eigenvalues based on sharp upper and lower bounds, together with a corresponding two-term counting function established using the geoesic billiards approach. Finally,we give an explicit bound on $p$ in terms of the dimension ensuring the corresponding lunes satisfy Pólya's conjecture for all eigenvalues.

math.SP↗

arXiv2024

Pólya-type inequalities on spheres and hemispheres

Given an eigenvalue $λ$ of the Laplace-Beltrami operator on $n-$spheres or $-$hemispheres, with multiplicity $m$ such that $λ=λ_{k}=\dots = λ_{k+m-1}$, we characterise the lowest and highest orders in the set $\left\{k,\dots,k+m-1\right\}$ for which Pólya's conjecture holds and fails. In particular, we show that Pólya's conjecture holds for hemispheres in the Neumann case, but not in the Dirichlet case when $n$ is greater than two. We further derive Pólya-type inequalities by adding a correction term providing sharp lower and upper bounds for all eigenvalues. This allows us to measure the deviation from the leading term in the Weyl asymptotics for eigenvalues on spheres and hemispheres. As a direct consequence, we obtain similar results for domains which tile hemispheres. We also obtain direct and reversed Li-Yau inequalities for $\mathbb{S}^2$ and $\mathbb{S}^4$, respectively.

math.SP↗

arXiv2023

Families of non-tiling domains satisfying Pólya's conjecture

We show the existence of classes of non-tiling domains satisfying Pólya's conjecture in any dimension, in both the Euclidean and non-Euclidean cases. This is a consequence of a more general observation asserting that if a domain satisfies Pólya's conjecture eventually, that is, for a sufficiently large order of the eigenvalues, and may be partitioned into $p$ non-overlapping isometric sub-domains, with $p$ arbitrarily large, then there exists an order $p_{0}$ such that for $p$ larger than $p_{0}$ all such sub-domains satisfy Pólya's conjecture. In particular, this allows us to show that families of sectors of domains of revolution with analytic boundary, and thin cylinders satisfy Pólya's conjecture, for instance. We also improve upon the Li-Yau constant for general cylinders in the Dirichlet case.

math.SP↗

arXiv2021

Grassman manifolds as subsets of Euclidean spaces

We consider the Grassman manifold $G(E)$ as the subset of all orthogonal projections of a given Euclidean space $E$ and obtain some explicit formulas concerning the differential geometry of $G(E)$ as a submanifold of $L(E,E)$ endowed with the Hilbert-Schmidt inner product. Most of these formulas can be naturally extended to the infinite dimensional Hilbert space case.

math.DG↗

arXiv2017

Dirichlet principal eigenvalue comparison theorems in geometry with torsion

We describe min-max formulas for the principal eigenvalue of a $V$-drift Laplacian defined by a vector field $V$ on a geodesic ball of a Riemannian manifold $N$. Then we derive comparison results for the principal eigenvalue with the one of a spherically symmetric model space endowed with a radial vector field, under pointwise comparison of the corresponding radial sectional and Ricci curvatures, and of the radial component of the vector fields. These results generalize the known case $V=0$.

math.DG↗

arXiv2010

A Spectral Bernstein Theorem

We study the spectrum of the Laplace operator of a complete minimal properly immersed hypersurface $M$ in $\R^{n+1}$. (1) Under a volume growth condition on extrinsic balls and a condition on the unit normal at infinity, we prove that $M$ has only essential spectrum consisting of the half line $[0, +\infty)$. This is the case when $\lim_{\tilde{r}\to +\infty}\tilde{r}κ_i=0$, where $\tilde{r}$ is the extrinsic distance to a point of $M$ and $κ_i$ are the principal curvatures. (2) If the $κ_i$ satisfy the decay conditions $|κ_i|\leq 1/\tilde{r}$, and strict inequality is achieved at some point $y\in M$, then there are no eigenvalues. We apply these results to minimal graphic and multigraphic hypersurfaces.

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arXiv2009

Self-Similar Solutions to Curvature Flow of Convex Hypersurfaces

We classify the self-similar solutions to a class of Weingarten curvature flow of connected compact convex hypersurfaces, isometrically immersed into space forms with non-positive curvature, and obtain a new characterization of a sphere in a Euclidean space $\R^{n+1}$.

math.DG↗

arXiv2007

Forced Convex Mean Curvature Flow in Euclidean Spaces

In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can also converge to a round sphere for some special forcing term and initial hypersurface. Furthermore, the normalization of the flow is carried out so that long time existence and convergence of the rescaled flow are studied. Our work extends Huisken's well-known mean curvature flow and McCoy's mixed volume preserving mean curvature flow.

math.DG↗