Upper bounds for the second nonzero eigenvalue of the Laplacian via folding and conformal volume

TL;DR

This paper establishes an upper bound for the second nonzero eigenvalue of the Laplacian on closed Riemannian manifolds, linking it to conformal volume, thus generalizing many existing results in spectral geometry.

Contribution

It introduces a new upper bound for the eigenvalue based on conformal volume, extending previous bounds to a broader class of manifolds.

Findings

Provides an effective upper bound for the eigenvalue on various manifolds

Generalizes known spectral bounds using conformal volume

Connects eigenvalue estimates with geometric conformal invariants

Abstract

We prove an upper bound for the volume-normalized second nonzero eigenvalue of the Laplace operator on closed Riemannian manifold, in terms of the conformal volume. This bound provides effective upper bound for a large class of manifolds, thereby generalizing many known results.