The PPW conjecture in curved spaces

In Euclidean ([1]) and Hyperbolic ([5]) space, and the round hemisphere ([2]), geodesic balls maximize the gap λ2−λ1 of Dirichlet eigenvalues, among domains with fixed λ1. We prove an upper bound on λ2−λ1 for domains in manifolds with certain curvature bounds. The inequality is sharp on geodesic balls in spaceforms.