Eigenvalue pinching on convex domains in space forms
  In this paper, we show that the convex domains of the hyperbolic space which are almost extremal for the Faber-Krahn or the Payne-Polya-Weinberger inequalities are close to geodesic balls. Our proof is also valid in other space forms and allows us to recover known results in Euclidean space and on the sphere. 
E. Aubry, J.Bertrand, B. Colbois