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 R n+1 , n ≤ 5, with constant mean curvature H = 0, provided that, for suitable p, the L p -norm of the traceless part of second fundamental form satisfies some growth condition.

On the entropies of hypersurfaces with bounded mean curvature

“…In this section we recall some results obtained in [28] and needed in the sequel. We assume that N is an orientable Riemannian manifold with bounded sectional curvature.…”

“…ϕ := |A − Hg|. In the present article we will need the following Caccioppoli's inequalities (Theorems 5.1 and 5.4 of [28]) and a reverse Hölder inequality (Theorem 4.2 [28]).…”

“…It is worthwhile to note that, if M is stable, using a suitable test function in inequality (58) of the proof of Theorem 4.2 in [28], one can deduce…”

“…This gives, in particular, a comparison between the extrinsic volume entropy of the hypersurface and the (intrinsic) volume entropy of the ambient space. In order to relate the volume entropy to the total curvature entropy (see Section 5 for the Definition) of finite index hypersurfaces with constant mean curvature we use some Cacciopoli's inequalities (obtained in [28]). Finally, using these results about entropies, we obtain some relations between spectra and curvature and various nonexistence results concerning some classes of hypersurfaces, including those of constant mean curvature stable or of finite index.…”

“…The construction of embedded half-tubes around a complete hypersurface with bounded curvature and mean curvature bounded away from zero, that is properly embedded in a simply-connected manifold of bounded curvature, is carried out in Section 3. Some results from [28] about Caccioppoli's inequalities for constant mean curvature hypersurfaces of finite index are recalled in Section 4. Next, in Section 5, various notions of entropies are introduced and discussed.…”