Voici les éléments 1 - 10 sur 73
- PublicationMétadonnées seulement
- PublicationAccès libreTwo properties of volume growth entropy in Hilbert geometry(2014-11-9)
;Verovic, PatrickThe aim of this paper is to provide two examples in Hilbert geometry which show that volume growth entropy is not always a limit on the one hand, and that it may vanish for a non-polygonal domain in the plane on the other hand.
- PublicationAccès libreCompact manifolds with fixed boundary and large Steklov eigenvaluesLet $(M,g)$ be a compact Riemannian manifold with boundary. Let $b>0$ be the number of connected components of its boundary. For manifolds of dimension $\geq 3$, we prove that for $j=b+1$ it is possible to obtain an arbitrarily large Steklov eigenvalue $\sigma_j(M,e^\delta g)$ using a conformal perturbation $\delta\in C^\infty(M)$ which is supported in a thin neighbourhood of the boundary, with $\delta=0$ on the boundary. For $j\leq b$, it is also possible to obtain arbitrarily large eigenvalues, but the conformal factor must spread throughout the interior of $M$. This is in stark contrast with the situation for the eigenvalues of the Laplace operator, for which the supremum is bounded in each fixed conformal class. In fact, when working in a fixed conformal class, it is known that the volume of $(M,e^\delta g)$ has to tend to infinity in order for some $\sigma_j$ to become arbitrarily large. We also prove that it is possible to obtain large eigenvalues while keeping different boundary components arbitrarily close to each others, by constructing a convenient Riemannian submersion.
- PublicationMétadonnées seulementIsoperimetric Inequalities for the Magnetic Neumann and Steklov Problems with Aharonov–Bohm Magnetic Potential(2022-9-14)
; ;Provenzano, LuigiSavo, Alessandro
- PublicationMétadonnées seulementA pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space(2007)
;Grosjean, Jean-FrançoisIn this paper, we give pinching theorems for the first nonzero eigenvalue lambda(1) (M) of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of M is I then, for any epsilon > 0, there exists a constant C, depending on the dimension n of M and the L-infinity-norm of the mean curvature H, so that if the L-2p-norm parallel to H parallel to(2p) (p >= 2) of H satisfies n parallel to H parallel to(2)(2p)-C-epsilon < lambda(1) (M), then the Hausdorff-distance between M and a round sphere of radius (n/lambda(1) (M))(1/2) is smaller than epsilon. Furthermore, we prove that if C is a small enough constant depending on n and the L-infinity-norm of the second fundamental form, then the pinching condition n parallel to H parallel to(2)(2p)-C < lambda(1) (M) implies that M is diffeomorphic to an n-dimensional sphere.
- PublicationMétadonnées seulementUniform stability of the Dirichlet spectrum for rough perturbations
- PublicationMétadonnées seulementEigenvalue control for a Finsler--Laplace operator(2013-5-1)
- PublicationAccès libreHypersurfaces with prescribed boundary and small Steklov eigenvaluesiven a smooth compact hypersurface $M$ with boundary $\Sigma=\partial M$, we prove the existence of a sequence $M_j$ of hypersurfaces with the same boundary as $M$, such that each Steklov eigenvalue $\sigma_k(M_j)$ tends to zero as $j$ tends to infinity. The hypersurfaces $M_j$ are obtained from $M$ by a local perturbation near a point of its boundary. Their volumes and diameters are arbitrarily close to those of $M$, while the principal curvatures of the boundary remain unchanged.
- PublicationMétadonnées seulementArea of ideal triangles and Gromov hyperbolicity in Hilbert Geometry(2008)
; ;Vernicos, ConstantinVerovic, Patrick