Optimal Estimates for Steklov Eigenvalue Gaps and Ratios on Warped Product Manifolds

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Abstract We consider an $n$-dimensional smooth Riemannian manifold $M^n=[0,R)\times \mathbb{S}^{n-1}$ endowed with a warped product metric $g=dr^2+h^2(r)g_{\mathbb{S}^{n-1}}$ and diffeomorphic to a Euclidean ball. Suppose that $M$ has strictly convex boundary. First, for the classical Steklov eigenvalue problem, we derive an optimal lower (upper, respectively) bound for its eigenvalue gaps in terms of $h^{\prime}(R)/h(R)$ when $n\geq 2$ and $Ric_g\geq 0$ ($\leq 0$, respectively). Second, in the same spirit, for two 4th-order Steklov eigenvalue problems studied by Kuttler and Sigillito in 1968, we deduce an optimal lower bound for their eigenvalue gaps in terms of either $h^{\prime}(R)/h^3(R)$ or $h^{\prime}(R)/h(R)$ when $n=2$ and the Gaussian curvature is nonnegative. We also consider optimal estimates on the eigenvalue ratios for these eigenvalue problems.