For Witten-Laplacians with radial log-concave measures on space forms, the geodesic ball uniquely minimizes the harmonic mean of the first n nonzero Neumann eigenvalues.
Benguria, Barbara Brandolini, and Francesco Chiacchio
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An isoperimetric inequality for Neumann eigenvalues with radial log-concave measures
For Witten-Laplacians with radial log-concave measures on space forms, the geodesic ball uniquely minimizes the harmonic mean of the first n nonzero Neumann eigenvalues.