A comparison theorem for the first non-zero Steklov eigenvalue
Let \(B_r(P)\) be the geodesic ball of radius \(r\) with center at \(P\in M^n\), where \(M^n\) is a complete Riemannian manifold and where \(r\) is less than the injectivity radius. Consider the problem: \(\Delta \phi=0\) on \(B_r(P)\) and \(\partial_n\phi=\nu\phi\) on \(bd(B_r(P))\) where \(\nu\) is a real number and \(\partial_n\) is the normal. This problem is called the Steklov problem. Let \(\nu_1\) be the first non-zero eigenvalue. Let \(B_r^{K_0}\) be the corresponding geodesic ball of radius \(r\) in a simply connected space form of constant sectional curvature \(K_0\). The author establishes the following estimate: Theorem. Suppose \(n=2,3\) and that the sectional curvatures \(\kappa(\pi)\leq K_0\) for any \(2\) plane \(\pi\) in the tangent space at the boundary which contain the normal vector field \(\partial_n\). Let \(r\) be less than the injectivity radius. Then \(\nu_1(B_r(P))\leq\nu_1(B_r^{K_0})\). Equality holds only when \(B_r(P)\) is isometric to \(B_r^{K_0}\). The author also shows that the assumption on the injective radius is necessary and establishes a similar estimate in higher dimensions.
scientific article; zbMATH DE number 1563422