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Caccioppoli's inequalities on constant mean curvature hypersurfaces in Riemannian manifolds

Abstract : We prove some Caccioppoli's inequalities for the traceless part of the second fundamental form of a complete, noncompact, finite index, constant mean curvature hypersurface of a Riemannian manifold, satisfying some curvature conditions. This allows us to unify and clarify many results scattered in the literature and to obtain some new results. For example, we prove that there is no stable, complete, noncompact hypersurface in Rn+1, n ≤ 5, with constant mean curvature H 6= 0, provided that, for suitable p, the Lp-norm of the traceless part of second fundamental form satisfies some growth condition.
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https://hal.archives-ouvertes.fr/hal-00849270
Contributor : Marc Soret <>
Submitted on : Tuesday, July 30, 2013 - 3:19:54 PM
Last modification on : Saturday, June 9, 2018 - 1:17:17 AM

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  • HAL Id : hal-00849270, version 1

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Marc Soret, Said Ilias, Barbara Nelli, Marc Soret, Marc Soret. Caccioppoli's inequalities on constant mean curvature hypersurfaces in Riemannian manifolds. Ann. Global Anal. Geom., 2012, 42 (4), pp.443-471. ⟨hal-00849270⟩

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