  Let M be a compact Riemannian manifold with boundary. Let b>0 be the number
of connected components of its boundary. For manifolds of dimension at least 3,
we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov
eigenvalue using a smooth conformal perturbation which is supported in a thin
neighbourhood of the boundary, identically equal to 1 on the boundary. For
j<b+1, it is also possible to obtain arbitrarily large j-th eigenvalue, but
this require the conformal factor to spread throughout the interior of the
manifold M. This is in stark contrast with the situation for the eigenvalues of
the Laplace operator on a closed manifold, where a conformal factor that is
large enough for the volume to become unbounded results in the spectrum
collapsing to 0. We also prove that it is possible to obtain large Steklov
eigenvalues while keeping different boundary components arbitrarily close to
each other, by constructing a convenient Riemannian submersion.
