Multiple tubular excisions and large Steklov eigenvalues

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.