Bounding the eigenvalues of the Laplace-Beltrami operator on compact submanifolds

TL;DR

This paper establishes upper bounds for the eigenvalues of the Laplace-Beltrami operator on compact submanifolds, depending on geometric and intersection properties, and shows limitations in controlling the first eigenvalue solely by volume and dimension.

Contribution

It introduces new eigenvalue bounds based on intersection counts and volume concentration, which are asymptotically optimal, and demonstrates limitations in controlling the first eigenvalue for hypersurfaces.

Findings

Eigenvalue bounds depend on intersection points and volume concentration.

Bounds are asymptotically optimal according to Weyl law.

First eigenvalue cannot be controlled solely by volume and dimension for hypersurfaces.

Abstract

We give upper bounds for the eigenvalues of the La-place-Beltrami operator of a compact $m$-dimensional submanifold $M$ of $\R^{m+p}$. Besides the dimension and the volume of the submanifold and the order of the eigenvalue, these bounds depend on either the maximal number of intersection points of $M$ with a $p$-plane in a generic position (transverse to $M$), or an invariant which measures the concentration of the volume of $M$ in $\R^{m+p}$. These bounds are asymptotically optimal in the sense of the Weyl law. On the other hand, we show that even for hypersurfaces (i.e., when $p=1$), the first positive eigenvalue cannot be controlled only in terms of the volume, the dimension and (for $m\ge 3$) the differential structure.