A characterization of the disk by eigenfunction of the Steklov eigenvalue

Abstract

Suppose M is homeomorphic to 2-dimensional sphere and let $f_i$, $1\le i\le 3$, be the first eigenfunctions of the Laplacian on M. Cheng proved (Proc Am Math Soc 55(2):379–381, 1976) that if $\sum _{i=1}^3 f_i^2$ is a constant, then M is isometric to a sphere of constant curvature. In view of the similarities between eigenvalues of the Laplacian and Steklov eigenvalue, we study eigenfuction of the first nonzero Steklov eigenvalue of a 2-dimensional compact manifold with boundary $\partial M$. Suppose M is a domain equipped with the flat metric g, and let f be an eigenfunction of the first nonzero Steklov’s eigenvalue. We prove that if $\nabla _g f$ is parallel along $\partial M$, the Lie bracket of the vector field orthogonal to $\nabla _g f$ and the tangent to $\partial M$ is zero, and $|\nabla _g f|^2$ is constant in M, then (M, g) is isometric to the disk equipped with the flat metric. We also prove that if the eigenfunctions $f_1, f_2$ corresponding to the first nonzero Steklov eigenvalue satisfy that $f_1^2+f_2^2$ is constant on $\partial M$ and $\varphi _*(g)$ is a Riemannian metric conformal to the flat metric of $D^2$ where $\varphi =(f_1,f_2)$, then (M, g) is isometric to the disk equipped with the flat metric. In another direction, we prove that a simply connected domain in the hyperbolic space $\mathbb {H}^3$ such that its Steklov eigenvalues are the same as a geodesic ball must be isometric to the geodesic ball.