Girouard, Alexandre

Nom

Girouard, Alexandre

Affiliation principale

Institut de mathématiques

Identifiants

https://libra.unine.ch/handle/123456789/464

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Accès libre
Compact manifolds with fixed boundary and large Steklov eigenvalues
(2019-8-22)
Colbois, Bruno
;
El Soufi, Ahmad
;
Girouard, Alexandre
Let $(M,g)$ 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.