Generic properties of Steklov eigenfunctions

TL;DR

This paper proves that for generic smooth metrics on compact manifolds with boundary, Steklov eigenvalues are simple and eigenfunctions are Morse functions, extending classical Laplacian results to the Steklov problem.

Contribution

It establishes generic simplicity of Steklov eigenvalues and Morse properties of eigenfunctions on manifolds with boundary, generalizing Uhlenbeck's results.

Findings

Nonzero Steklov eigenvalues are simple for generic metrics.

Steklov eigenfunctions have zero as a regular value.

Eigenfunctions are Morse functions on the boundary.

Abstract

Let $M^n$ be a smooth compact manifolds with smooth boundary. We show that for a generic $C^k$ metic on $\bar{M^n}$ 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.