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Caccioppoli's inequalities on constant mean curvature hypersurfaces in Riemannian manifolds
Differential Geometry · Submitted 19 Jul 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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This is a revised version (minor changes and a deeper insight in the positive curvature case). 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 ${\mathbb R}^{n+1},$ $n\leq 5,$ with constant mean curvature $H\not=0,$ provided that, for suitable $p,$ the $L^p$-norm of the traceless part of second fundamental form satisfies some growth condition.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This is a revised version (minor changes and a deeper insight in the positive curvature case). | background 0.78 | 0.36 | 0.02 | asserted 0.89 |
| s1 | 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. | contribution 0.74 | 0.96 | 0.88 | asserted 1.00 |
| s2 | This allows us to unify and clarify many results scattered in the literature and to obtain some new results. | interpretation 0.75 | 0.88 | 0.12 | asserted 0.97 |
| s3 | For example, we prove that there is no stable, complete, noncompact hypersurface in ${\mathbb R}^{n+1},$ $n\leq 5,$ with constant mean curvature $H\not=0,$ provided that, for suitable $p,$ the $L^p$-norm of the traceless part of second fundamental form satisfies some growth condition. | result 0.99 | 0.95 | 0.26 | asserted 1.00 |
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Original abstract
This is a revised version (minor changes and a deeper insight in the positive curvature case). 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 ${\mathbb R}^{n+1},$ $n\leq 5,$ with constant mean curvature $H\not=0,$ provided that, for suitable $p,$ the $L^p$-norm of the traceless part of second fundamental form satisfies some growth condition.
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Rosenberg [ref] proved a formula analogous to the result of Corollary [ref] for surfaces with parallel mean curvature.
Surfaces with parallel mean curvature in $\mathbb{S}^3\times\mathbb{R}$ and $\mathbb{H}^3\times\mathbb{R}$ · Simons' inequality for constant mean curvature hypersurfaces · 0.87
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Batista [ref] proved a formula analogous to the result of Corollary [ref] for surfaces in $ℍ^2×ℝ$ and $𝕊^2×ℝ.$
Simons' Type Equation in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$ and Applications · Simons' inequality for constant mean curvature hypersurfaces · 0.94
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