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Compact manifolds with fixed boundary and large Steklov eigenvalues

Authors: Bruno Colbois, Ahmad El Soufi and Alexandre Girouard
Journal: Proc. Amer. Math. Soc. 147 (2019), 3813-3827
MSC (2010): Primary 35P15, 53C21, 58J50; Secondary 53C23, 53C20
Published electronically: May 17, 2019
MathSciNet review: 3993774
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Abstract: 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 $M$. In fact, when working in a fixed conformal class and for $\delta =0$ on the boundary, 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. This is in stark contrast with the situation for the eigenvalues of the Laplace operator on a closed manifold, where a conformal factor that is large enough for the volume to become unbounded results in the spectrum collapsing to 0. We also prove that it is possible to obtain large Steklov eigenvalues while keeping different boundary components arbitrarily close to each other, by constructing a convenient Riemannian submersion.

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Additional Information

Bruno Colbois
Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Emile-Argand 11, CH-2000 Neuchâtel, Switzerland
MR Author ID: 50460

Ahmad El Soufi
Affiliation: Laboratoire de Mathématiques et Physique Théorique, UMR-CNRS 6083, Université François Rabelais de Tours, Parc de Grandmont, 37200 Tours, France

Alexandre Girouard
Affiliation: Département de mathématiques et de statistique, Pavillon Alexandre-Vachon, Université Laval, Québec, Québec, G1V 0A6, Canada
MR Author ID: 832728

Received by editor(s): February 2, 2017
Received by editor(s) in revised form: October 14, 2018
Published electronically: May 17, 2019
Additional Notes: During the first week of 2017, the first and third authors were supposed to travel to Tours and work with Ahmad El Soufi to complete this paper. We learned just a few days before our visit of his untimely death. Ahmad was a colleague and a friend. He will be dearly missed.
Communicated by: Michael Wolf
Article copyright: © Copyright 2019 American Mathematical Society