Multiple tubular excisions and large Steklov eigenvalues

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