Some results on higher eigenvalue optimization

In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for $k\geq 3$. For $k=1$ the classical result of [W] shows that $\sigma_1$ is maximized by the standard metric on the round disk. For $k=2$ it was shown [GP1] that $\sigma_2$ is not maximized for a smooth metric. We also prove a local rigidity result for the critical catenoid and the critical M\"obius band as free boundary minimal surfaces in a ball under $C^2$ deformations. We next show that the first $k$ Steklov eigenvalues are continuous under certain degenerations of Riemannian manifolds in any dimension. Finally we show that for $k\geq 2$ the supremum of the $k$-th Steklov eigenvalue on the annulus over all metrics is strictly larger that that over $S^1$-invariant metrics. We prove this same result for metrics on the M\"obius band.