A lower bound for first eigenvalue with mixed boundary condition

Abstract

Let M be an n-dimensional compact Riemannian manifold with or without boundary, and its Ricci curvature Ric M⩾n−1. The paper obtains an inequality for the first eigenvalue η 1 of M with mixed boundary condition, which is a generalization of the results of Lichnerowicz, Reilly, Escobar and Xia. It is also proved that η 1⩾n for certain n-dimensional compact Riemannian manifolds with boundary, which is an extension of the work of Cheng, Li and Yau.