A characterization of the disk by eigenfunction of the Steklov eigenvalue

作者：Pak Tung Ho

来源：Manuscripta Mathematica, 2017, 154(3-4): 297-307.

DOI：10.1007/s00229-017-0916-9

Suppose M is homeomorphic to 2-dimensional sphere and let , , be the first eigenfunctions of the Laplacian on M. Cheng proved (Proc Am Math Soc 55(2):379-381, 1976) that if 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 . 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 is parallel along , the Lie bracket of the vector field orthogonal to and the tangent to is zero, and is constant in M, then (M, g) is isometric to the disk equipped with the flat metric. We also prove that if the eigenfunctions corresponding to the first nonzero Steklov eigenvalue satisfy that is constant on and is a Riemannian metric conformal to the flat metric of where , 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 such that its Steklov eigenvalues are the same as a geodesic ball must be isometric to the geodesic ball.

• 出版日期2017-11