Generic properties of Steklov eigenfunctions

Abstract

Let M n M^n be a smooth compact manifold with smooth boundary. We show that for a generic C k C^k metric on M n ¯ {}\mkern 3mu\overline {\mkern -3muM^n} with k > n − 1 k>n-1 , the non-zero Steklov eigenvalues are simple. Moreover, we 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.