  Using methods in the spirit of deterministic homogenisation theory we obtain
convergence of the Steklov eigenvalues of a sequence of domains in a Riemannian
manifold to weighted Laplace eigenvalues of that manifold. The domains are
obtained by removing small geodesic balls that are asymptotically densely
uniformly distributed as their radius tends to zero. We use this relationship
to construct manifolds that have large Steklov eigenvalues.
  In dimension two, and with constant weight equal to 1, we prove that
Kokarev's upper bound of $8\pi$ for the first nonzero normalised Steklov
eigenvalue on orientable surfaces of genus 0 is saturated. For other
topological types and eigenvalue indices, we also obtain lower bounds on the
best upper bound for the eigenvalue in terms of Laplace maximisers. For the
first two eigenvalues, these lower bounds become equalities. A surprising
consequence is the existence of free boundary minimal surfaces immersed in the
unit ball by first Steklov eigenfunctions and with area strictly larger than
$2\pi$. This was previously thought to be impossible. We provide numerical
evidence that some of the already known examples of free boundary minimal
surfaces have these properties and also exhibit simulations of new free
boundary minimal surfaces of genus 0 in the unit ball with even larger area.
The first nonzero Steklov eigenvalue of all these examples is equal to 1, as a
consequence of their symmetries and topology, so that they verify a general
conjecture by Fraser and Li.
  In dimension three and larger, we prove that the isoperimetric inequality of
Colbois--El Soufi--Girouard is sharp and implies an upper bound for weighted
Laplace eigenvalues. We also show that in any manifold with a fixed metric, one
can construct by varying the weight a domain with connected boundary whose
first nonzero normalised Steklov eigenvalue is arbitrarily large.
