On a Conjecture of Ashbaugh and Benguria about Lower Eigenvalues of the Neumann Laplacian

In this paper, we prove an isoperimetric inequality for lower order eigenvalues of the free membrane problem on bounded domains of a Euclidean space or a hyperbolic space which strengthens the well-known Szeg"o-Weinberger inequality and supports strongly an important conjecture of Ashbaugh-Benguria.