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.