An Overdetermined Steklov Eigenvalue Problem on Riemannian Manifolds with Nonnegative Ricci Curvature

We consider an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature. We prove that, given a compact connected domain $$\Omega $$ with nonnegative Gaussian curvature with $$C^2$$ boundary, if the first Steklov eigenfunction is a solution to the overdetermined problem, then the domain $$\Omega $$ is flat and the boundary $$\partial \Omega $$ consists of geodesics or geodesic circles. This can be regarded as a generalization of the result by Payne-Philippin [21], where they assumed that $$\Omega $$ is a simply-connected domain in $$\mathbb {R}^2$$ . We also obtain a similar result for the same overdetermined problem on a higher-dimensional compact connected manifold with $$C^2$$ boundary with nonnegative Ricci curvature.