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

TL;DR

This paper establishes Caccioppoli's inequalities for constant mean curvature hypersurfaces in Riemannian manifolds, providing unified results and new insights, especially in the positive curvature case, with implications for stability and geometric analysis.

Contribution

It introduces new Caccioppoli's inequalities for hypersurfaces with constant mean curvature, unifying existing results and deriving novel nonexistence theorems under curvature and growth conditions.

Findings

No stable, complete, noncompact hypersurfaces in R^{n+1} with nonzero constant mean curvature for n ≤ 5 under certain growth conditions.

Unified framework for inequalities related to the second fundamental form.

New nonexistence results for hypersurfaces with specific curvature and stability properties.

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.