Some eigenvalue estimates on Riemannian manifolds

Abstract: The eigenvalues of the Laplacian on a Riemannian manifold and the geometry of the manifold are related. In particular, the first nonzero eigenvalue of the Laplacian plays an important role in Riemannian geometry and finding some bounds of this eigenvalue is a fundamental question. In this talk, we will focus on eigenvalue estimates of certain eigenvalue problems.

We will begin by considering Steklov eigenvalue problem on a star-shaped bounded domain in the unit n-sphere and a paraboloid, and give a sharp lower bound for the Steklov eigenvalues. We will also provide a two sided sharp bound for all Steklov eigenvalues on a ball with rotationally invariant metric and with bounded radial curvature.

Next we will discuss the mixed Steklov-Dirichlet problem and Neumann eigenvalue problem on a class of doubly-connected domains in Rⁿ(n > 2) and rank-1 symmetric space, respectively, and find domains which maximize the first nonzero eigenvalue of these problems.