The PPW conjecture in curved spaces

TL;DR

This paper extends the understanding of eigenvalue gaps by establishing an upper bound for domains in curved spaces, generalizing known results from Euclidean and hyperbolic geometries, with sharpness demonstrated on spaceform geodesic balls.

Contribution

It proves a new upper bound on the eigenvalue gap for domains in manifolds with curvature bounds, generalizing previous results from Euclidean and hyperbolic spaces.

Findings

Upper bound on eigenvalue gap in curved spaces

Sharpness of inequality on geodesic balls in spaceforms

Generalization of Euclidean and hyperbolic results

Abstract

In Euclidean and Hyperbolic space, and the hemisphere in $S^n$, geodesic balls maximize the gap $\lambda_2 - \lambda_1$ of Dirichlet eigenvalues, amoung domains with fixed $\lambda_1$. We prove an upper bound on $\lambda_2 - \lambda_1$ for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms.