Generic properties of Steklov eigenfunctions

Abstract

Let $M^n$ be a smooth compact manifolds with smooth boundary. We show that for a generic $C^k$ metic on $\overline{M^n}$ with $k>n-1$, the nonzero Steklov eigenvalues are simple. Moreover, we also prove that the non-constant Steklov eigenfunctions have zero as a regular value and are Morse functions on the boundary for such generic metric. These results generalize the celebrated results on Laplacians by Uhlenbeck to the Steklov setting.

to be published in the Transactions of the American Mathematical Society