Multiple tubular excisions and large Steklov eigenvalues

Abstract

Given a closed Riemannian manifold M and $b\ge 2$ closed connected submanifolds $N_j\subset M$ of codimension at least 2, we prove that the first nonzero eigenvalue of the domain $\Omega _\varepsilon \subset M$ obtained by removing the tubular neighbourhood of size $\varepsilon$ around each $N_j$ tends to infinity as $\varepsilon$ tends to 0. More precisely, we prove a lower bound in terms of $\varepsilon$, b, the geometry of M and the codimensions and the volumes of the submanifolds and an upper bound in terms of $\varepsilon$ and the codimensions of the submanifolds. For eigenvalues of index $k=b,b+1,\ldots$, we have a stronger result: their order of divergence is $\varepsilon ^{-1}$ and their rate of divergence is only depending on m and on the codimensions of the submanifolds.