  Given a closed manifold $M$ and a closed connected submanifold $N\subset M$
of positive codimension, we study the Steklov spectrum of the domain
$\Omega_\varepsilon\subset M$ obtained by removing the tubular neighbourhood of
size $\varepsilon$ around $N$. All non-zero eigenvalues in the mid-frequency
range tend to infinity at a rate which depends only on the codimension of $N$
in $M$. Eigenvalues above the mid-frequency range are also described: they tend
to infinity following an unbounded sequence of clusters. This construction is
then applied to obtain manifolds with unbounded perimeter-normalized spectral
gap and to show the necessity of using the injectivity radius in some known
isoperimetric-type upper bounds.
