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 non-zero eigenvalue of the domain ${\mathrm{\Omega }}_{\epsilon }\subset M$ obtained by removing the tubular neighbourhood of size $\epsilon$ around each $N_j$ tends to infinity as $\epsilon$ tends to $0$. More precisely, we prove a lower bound in terms of $\epsilon$, $b$, the geometry of $M$ and the codimensions and the volumes of the submanifolds and an upper bound in terms of $\epsilon$ and the codimensions of the submanifolds. For eigenvalues of index $k=b,b+1,\dots$, we have a stronger result: their order of divergence is ${\epsilon }^{-1}$ and their rate of divergence is only depending on $m$ and on the codimensions of the submanifolds.