Conformal volume and eigenvalue problems

TL;DR

This paper extends classical inequalities relating Laplace eigenvalues to geometric quantities on Riemannian manifolds, providing new bounds for higher eigenvalues, negative eigenvalues of Schrödinger operators, and minimal hypersurfaces.

Contribution

It generalizes classical eigenvalue inequalities to higher eigenvalues and derives bounds for Schrödinger operators and minimal hypersurfaces.

Findings

Inequalities for higher Laplace eigenvalues in terms of conformal volume.

Bounds for the number of negative eigenvalues of Schrödinger operators.

Index bounds for minimal hypersurfaces in spheres.

Abstract

We prove inequalities for Laplace eigenvalues on Riemannian manifolds generalising to higher eigenvalues two classical inequalities for the first Laplace eigenvalue - the inequality in terms of the $L^2$-norm of mean curvature, due to Reilly in 1977, and the inequality in terms of conformal volume, due to Li and Yau in 1982, and El Soufi and Ilias in 1986. We also obtain bounds for the number of negative eigenvalues of Schr\"odinger operators, and in particular, index bounds for minimal hypersurfaces in spheres.