On Cheng's Eigenvalue Comparison Theorems

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dc.creatorBessa, G. Pacelli
dc.creatorMontenegro, J. Fabio
dc.date2005-07-15
dc.date.accessioned2026-07-07T10:13:42Z
dc.date.available2026-07-07T10:13:42Z
dc.descriptionWe prove Cheng's eigenvalue comparison theorems for geodesic balls within the cut locus under weaker geometric hypothesis, and we also show that there are certain geometric rigidity in case of equality of the eigenvalues. This rigidity becomes isometric rigidity under upper sectional curvature bounds or lower Ricci curvature bounds. We construct examples of smooth metrics showing that our results are true extensions of Cheng's theorem. We also construct a family of complete smooth metrics on the Euclidean space non-isometric to the constant sectional curvature k metrics of the simply connected space forms of constant sectional curvature k such that the geodesic balls of radius r have the same first eigenvalue and the geodesic spheres have the same mean curvatures. In the end we construct examples of Riemannian manifolds M with arbitrary topology with positive fundamental tone positive that generalize Veeravalli's examples.
dc.descriptionpaper with 11 pages
dc.identifierhttps://arxiv.org/abs/math/0507318
dc.identifierhttp://arxiv.org/abs/math/0507318
dc.identifierMathematical Proceedings of the Cambridge Philosophical Society, v. 144, p. 673-682, 2008.
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/172624
dc.subjectDifferential Geometry
dc.subjectSpectral Theory
dc.subject53C40, 53C42, 58C40
dc.titleOn Cheng's Eigenvalue Comparison Theorems
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