Deformations of balls in schiffer's conjecture for riemannian symmetric spaces

It is proved that geodesic balls in a Riemannian symmetric space of rank one are stable solutions to a free-boundary problem for the Laplace -Beltrami operator with constant Dirichlet - Neumann boundary conditions. This result supports Schiffer's conjecture that balls are the only solutions to the problem. The main ingredient of the proof is a characterization of geodesic balls by the multiplicity of eigenvalues of the Laplace - Beltrami operator.