On the Spectra of Three Steklov Eigenvalue Problems on Warped Product Manifolds

Abstract

Let \(M^n=[0,R]\times \mathbb {S}^{n-1}\) be an n-dimensional (\(n\ge 2\)) smooth Riemannian manifold equipped with the warped product metric \(g=dr^2+h^2(r)g_{\mathbb {S}^{n-1}}\) and diffeomorphic to a Euclidean ball. Assume that M has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we obtain an optimal lower (upper, respectively) bound for its spectrum in terms of \(h'(R)/h(R)\) when \(\mathrm {Ric}_g\ge 0\) (\(\le 0\), respectively). Second, for two fourth-order Steklov eigenvalue problems studied by Kuttler and Sigillito in 1968, we derive a lower bound for their spectra in terms of either \(h'(R)/h^3(R)\) or \(h'(R)/h(R)\) when \(\mathrm {Ric}_g\ge 0\), which is optimal for certain cases; in particular, we confirm a conjecture raised by Q. Wang and C. Xia for warped product manifolds of dimension \(n=2\) or \(n\ge 4\). For some proofs we utilize the Reilly’s formula and reveal a new feature on its use.