A sharp lower bound for the first (nonzero) Steklov eigenvalue

Abstract: Escobar has conjectured that for a compact manifold with boundary which has nonnegative Ricci curvature and boundary principal curvatures bounded below by 1, the first (nonzero) Steklov eigenvalue is greater than or equal to 1，with equality holding only on a Euclidean ball. This conjecture is true in two dimensions due to Payne and Escobar. In this talk, we present a resolution to this conjecture in the case of nonnegative sectional curvature in any dimensions. We will also give a sharp comparison result between the first (nonzero) Steklov eigenvalue and the boundary first eigenvalue. Our tool is a weighted Reilly type formula due to Qiu-Xia and a Pohozaev type identity. The talk is based on a joint work with Changwei Xiong.

Mathematics

Audience: researchers in the topic

ICCM 2020