  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.
