Compact manifolds with fixed boundary and large Steklov eigenvalues

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.