The first nonzero eigenvalue of Neumann problem on Riemannian manifolds (Q1346085)

The author obtains some comparison theorems of the first non-zero Neumann eigenvalue on domains in a non-positively curved Riemannian manifold. The author first gives a generalized Szegö-Weinberger theorem (Theorem 1). Then the first non-zero Neumann eigenvalues for geodesic balls on non- positively curved Riemannian manifolds are compared (Theorem 2). Based on these results, a ``monotonicity principle'' for the Neumann eigenvalues is derived. Then the author proves a stability theorem of maximality of the first non-zero Neumann eigenvalue of a geodesic ball among those of all domains with the same volume.