Berger inequality for Riemannian manifolds with an upper sectional curvature bound

TL;DR

This paper extends Berger's inequality to all Laplace eigenvalues on Riemannian manifolds with an upper curvature bound, including conformal metrics and minimal submanifolds, with explicit eigenvalue estimates.

Contribution

It generalizes Berger's inequality to all eigenvalues and to broader geometric contexts, including conformal metrics and minimal submanifolds.

Findings

Inequalities for all Laplace eigenvalues under an upper curvature bound.

Extension of inequalities to conformal metrics.

Explicit eigenvalue estimates for minimal submanifolds.

Abstract

We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the first Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities continue to hold for conformal metrics, and moreover, extend naturally to minimal submanifolds. In addition, we obtain explicit estimates for Laplace eigenvalues of minimal submanifolds in terms of geometric quantities of the ambient space.