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 zero in the unit ball with even larger area. We prove that the first nonzero Steklov eigenvalue of all these examples is equal to 1, as a consequence of their symmetries and topology, so that they are consistent with 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.

使用确定性同质化理论的精神，我们获得了黎曼流形中域序列的 Steklov 特征值收敛到该流形的加权拉普拉斯特征值。这些域是通过去除小测地球获得的，这些小测地球随着半径趋于零而渐近密集均匀分布。我们使用这种关系来构造具有大 Steklov 特征值的流形。在第二维，恒重等于 1，我们证明 Kokarev 的上界\(8\pi \)对于属 0 的可定向表面上的第一个非零归一化 Steklov 特征值是饱和的。对于其他拓扑类型和特征值指数，我们还根据拉普拉斯最大值获得特征值的最佳上限的下限。对于前两个特征值，这些下限变为等式。一个令人惊讶的结果是存在自由边界最小表面，通过第一 Steklov 特征函数浸入单位球中，并且面积严格大于\(2\pi \). 这在以前被认为是不可能的。我们提供的数值证据表明，一些已知的自由边界最小曲面示例具有这些特性，并且还展示了对面积更大的单位球中的零属新自由边界最小曲面的模拟。我们证明所有这些例子的第一个非零 Steklov 特征值都等于 1，这是它们的对称性和拓扑结构的结果，因此它们与 Fraser 和 Li 的一般猜想是一致的。在第三维和更大的维度上，我们证明 Colbois-El Soufi-Giroard 的等周不等式是尖锐的，并且暗示了加权拉普拉斯特征值的上限。我们还表明，在任何具有固定度量的流形中，