  Given a closed Riemannian manifold $M$ and $b\geq2$ closed connected
submanifolds $N_j\subset M$ of codimension at least $2$, we prove that the
first non-zero 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.
