The PPW conjecture in curved spaces

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.

13 pages, final version