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Using methods in the spirit of deterministic homogenisation theory we obtainconvergence of the Steklov eigenvalues of a sequence of domains in a Riemannianmanifold to weighted Laplace eigenvalues of that manifold. The domains areobtained by removing small geodesic balls that are asymptotically denselyuniformly distributed as their radius tends to zero. We use this relationshipto construct manifolds that have large Steklov eigenvalues.  In dimension two, and with constant weight equal to 1, we prove thatKokarev's upper bound of $8\pi$ for the first nonzero normalised Stekloveigenvalue on orientable surfaces of genus 0 is saturated. For othertopological types and eigenvalue indices, we also obtain lower bounds on thebest upper bound for the eigenvalue in terms of Laplace maximisers. For thefirst two eigenvalues, these lower bounds become equalities. A surprisingconsequence is the existence of free boundary minimal surfaces immersed in theunit ball by first Steklov eigenfunctions and with area strictly larger than$2\pi$. This was previously thought to be impossible. We provide numericalevidence that some of the already known examples of free boundary minimalsurfaces have these properties and also exhibit simulations of new freeboundary 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 aconsequence of their symmetries and topology, so that they verify a generalconjecture by Fraser and Li.  In dimension three and larger, we prove that the isoperimetric inequality ofColbois--El Soufi--Girouard is sharp and implies an upper bound for weightedLaplace eigenvalues. We also show that in any manifold with a fixed metric, onecan construct by varying the weight a domain with connected boundary whosefirst nonzero normalised Steklov eigenvalue is arbitrarily large.

Large Steklov eigenvalues via homogenisation on manifolds