  Recently, D. Bucur and M. Nahon used boundary homogenisation to show the
remarkable flexibility of Steklov eigenvalues of planar domains. In the present
paper we extend their result to higher dimensions and to arbitrary manifolds
with boundary, even though in those cases the boundary does not generally
exhibit any periodic structure. Our arguments use framework of variational
eigenvalues and provides a different proof of the original results.
Furthermore, we present an application of this flexibility to the optimisation
of Steklov eigenvalues under perimeter constraint. It is proved that the best
upper bound for normalised Steklov eigenvalues of surfaces of genus zero and
any fixed number of boundary components can always be saturated by planar
domains. This is the case even though any actual maximiser (except for simply
connected surfaces) is always far from being planar themselves. In particular,
it yields sharp upper bound for the first Steklov eigenvalue of doubly
connected planar domains.
