Hilbert domains quasi-isometric to normed vector spaces
We prove that a Hilbert domain which is quasi-isometric to a normed vector space is actually a convex polytope.
math.MG↗
arXiv subjects
Bruno Colbois
Publications and source records attributed to Bruno Colbois.
Hilbert geometry for convex polygonal domains
We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to Euclidean plane.
math.MG↗
Eigenvalues estimate for the Neumann problem on bounded domains
In this note, we investigate upper bounds of the Neumann eigenvalue problem for the Laplacian of a bounded domain (with smooth boundary) in a given complete (not compact a priori) Riemannian manifold with Ricci bounded below . For this, we use test functions for the Rayleigh quotient subordinated to a family of open sets constructed in a general metric way, interesting for itself. As application, we get upper bounds for the Neumann spectrum which is clearly in agreement with the Weyl law and which is analogous to Buser's upper bounds of the spectrum of a closed Riemannian manifold with lower bound on the Ricci curvature.
math.DG↗
Some smooth Finsler deformations of hyperbolic surfaces
Given a closed hyperbolic Riemannian surface, the aim of the present paper is to describe an explicit construction of smooth deformations of the hyperbolic metric into Finsler metrics that are not Riemannian and whose properties are such that the classical Riemannian results about entropy rigidity, marked length spectrum rigidity and boundary rigidity all fail to extend to the Finsler category.
math.DG↗
Extremal $G$-invariant eigenvalues of the Laplacian of $G$-invariant metrics
The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere $S^2$ endowed with $S^1$-invariant metrics, we consider the subsequence $λ_k^G$ of the spectrum of a Riemannian manifold $M$ which corresponds to metrics and functions invariant under the action of a compact Lie group $G$. If $G$ has dimension at least 1, we show that the functional $λ_k^G$ admits no extremal metric under volume-preserving $G$-invariant deformations. If, moreover, $M$ has dimension at least three, then the functional $λ_k^G$ is unbounded when restricted to any conformal class of $G$-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on $S^n$; however, if we also require the metric to be induced by an embedding of $S^n$ in $\mathbb{R}^{n+1}$, we get an optimal upper bound on $λ_k^G$.
math.DG↗
A pinching theorem for the first eigenvalue of the laplacian on hypersurface of the euclidean space
In this paper, we give pinching Theorems for the first nonzero eigenvalue $λ$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is 1 then, for any $ε>0$, there exists a constant $C\_ε$ depending on the dimension $n$ of $M$ and the $L\_{\infty}$-norm of the mean curvature $H$, so that if the $L\_{2p}$-norm $\|H\|\_{2p}$ ($p\geq 2$) of $H$ satisfies $n\|H\|\_{2p}-C\_ε<λ$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/λ)^{1/2}$ is smaller than $ε$. Furthermore, we prove that if $C$ is a small enough constant depending on $n$ and the $L\_{\infty}$-norm of the second fundamental form, then the pinching condition $n\|H\|\_{2p}-C<\la$ implies that $M$ is diffeomorphic to an $n$-dimensional sphere.
math.DG↗
Les géométries de Hilbert sont à géométrie locale bornée
We prove that the Hilbert geometry of a convex domain in ${\mathbb R}^n$ has bounded local geometry, i.e., for a given radius, all balls are bilipschitz to a euclidean domain of ${\mathbb R}^n$. As a consequence, if the Hilbert geometry is also Gromov hyperbolic, then the bottom of its spectrum is strictly positive. We also give a counter exemple in dimension three which shows that the reciprocal is not true for non plane Hilbert geometries.
math.DG↗
Capacite et inegalite de Faber-Krahn dans l'espace euclidien
In this paper, we define a new capacity which allows us to control the behaviour of the Dirichlet spectrum of a compact Riemannian manifold with boundary, with "small" subsets (which may intersect the boundary) removed. This result generalizes a classical result of Rauch and Taylor ("the crushed ice theorem"). In the second part, we show that the Dirichlet spectrum of a sequence of bounded Euclidean domains converges to the spectrum of a ball with the same volume, if the first eigenvalue of these domains converges to the first eigenvalue of a ball.
math.DG↗
Bas du spectre et delta-hyperbolicité en géométrie de Hilbert plane
We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero
math.DG↗
Eigenvalues of the Laplacian acting on $p$-forms and metric conformal deformations
Let $(M,g)$ be a compact connected orientable Riemannian manifold of dimension $n\ge4$ and let $λ_{k,p} (g)$ be the $k$-th positive eigenvalue of the Laplacian $Δ_{g,p}=dd^*+d^*d$ acting on differential forms of degree $p$ on $M$. We prove that the metric $g$ can be conformally deformed to a metric $g'$, having the same volume as $g$, with arbitrarily large $λ_{1,p} (g')$ for all $p\in[2,n-2]$. Note that for the other values of $p$, that is $p=0, 1, n-1$ and $n$, one can deduce from the literature that, $\forall k >0$, the $k$-th eigenvalue $λ_{k,p}$ is uniformly bounded on any conformal class of metrics of fixed volume on $M$. For $p=1$, we show that, for any positive integer $N$, there exists a metric $g_N$ conformal to $g$ such that, $\forall k\le N$, $λ_{k,1} (g_N) =λ_{k,0} (g_N) $, that is, the first $N$ eigenforms of $Δ_{g_N,1}$ are all exact forms.
math.DG↗
Extremal eigenvalues of the Laplacian in a conformal class of metrics : the "conformal spectrum"
Let $M$ be a compact connected manifold of dimension $n$ endowed with a conformal class $C$ of Riemannian metrics of volume one. For any integer $k\geq0$, we consider the conformal invariant $λ_k ^c (C)$ defined as the supremum of the $k$-th eigenvalue $λ_k (g)$ of the Laplace-Beltrami operator $Δ_g$, where $g$ runs over $C$. First, we give a sharp universal lower bound for $λ_k ^c (C)$ extending to all $k$ a result obtained by Friedlander and Nadirashvili for $k=1$. Then, we show that the sequence $ \{λ_k ^c (C) \}$, that we call "conformal spectrum", is strictly increasing and satisfies, $\forall k\geq 0$, $λ_{k+1} ^c (C)^{n/2} - λ_k ^c (C)^{n/2} \geq n^{n/2} ω_n $, where $ω_n $ is the volume of the $n$-dimensional standard sphere. When $M$ is an orientable surface of genus $γ$, we also consider the supremum $λ_k ^{top} (γ)$ of $λ_k(g)$ over the set of all the area one Riemannian metrics on $M$, and study the behavior of $λ_k ^{top} (γ)$ in terms of $γ$.
math.DG↗
L'aire des triangles ideaux en geometrie de Hilbert
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the convex, which characterizes the triangle domains in the plane. Furthermore, under specific geometric assumptions, we obtain an upper bound, which depends on the convex domain.
math.DG↗