Eigenvalue estimates for a class of elliptic differential operators on compact manifolds

TL;DR

This paper derives eigenvalue estimates for a class of elliptic operators on compact manifolds, generalizing classical results like the Lichnerowicz-Obata Theorem, with applications to geometric analysis.

Contribution

It introduces a generalized Bochner formula for these operators and provides sharp eigenvalue bounds, extending known theorems in Riemannian geometry.

Findings

Sharp estimates for first nonzero eigenvalues

Generalized Bochner-type formula derived

Extensions of Lichnerowicz-Obata Theorem achieved

Abstract

The motivation of this paper is to study a second order elliptic operator which appears naturally in Riemannian geometry, for instance in the study of hypersurfaces with constant $r$-mean curvature. We prove a generalized Bochner-type formula for such a kind of operators and as applications we obtain some sharp estimates for the first nonzero eigenvalues in two special cases. These results can be considered as generalizations of the Lichnerowicz-Obata Theorem.