Abstract
In this article, the author obtained some comparison theorems of the first nonzero Neumann eigenvalue on domains in nonpositively curved Riemannian manifolds. The author first gives a generalized Szegö-Weinberger theorem (Theorem 1). Then the first nonzero Neumann eigenvalues for geodesic balls on nonpositively 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 nonzero Neumann eigenvalue of a geodesic ball among those of all domains with the same volume.
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Communicated by S.-Y. Cheng
Partially supported by the NSF under Grant DMS-9103184.
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Xu, Y. The first nonzero eigenvalue of Neumann problem on Riemannian manifolds. J Geom Anal 5, 151–165 (1995). https://doi.org/10.1007/BF02926446
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DOI: https://doi.org/10.1007/BF02926446