Bernstein-type theorems in hypersurfaces with constant mean curvature

Carmo, Manfredo P. do; Zhou, Detang

doi:10.1590/s0001-37652000000300003

Cited by 12 publications

(4 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…• Taking x = 1 in (1) of Corollary 6.3, we improve the result by H. Alencar and M. do Carmo, stated in Theorem 4 of [2]. Furthermore, M. do Carmo and D. Zhou [25] stated a result weaker than (1) of Corollary 6.3 and, in their proof, they use wrongly Young's inequality (see equation (3.7) there). Remark 6.4.…”

Section: Applications Of the Caccioppoli's Inequalities In The Stablementioning

confidence: 53%

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

Ilias¹,

Nelli²,

Soret³

2012

Ann Glob Anal Geom

View full text Add to dashboard Cite

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.

show abstract

Section: Applications Of the Caccioppoli's Inequalities In The Stablementioning

confidence: 53%

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

Ilias¹,

Nelli²,

Soret³

2012

Ann Glob Anal Geom

View full text Add to dashboard Cite

show abstract

“…Then, one uses a Liouville type theorem [63], to prove that such function does not exist. Many authors try to obtain a Bernstein type theorem for (strongly) stable constant mean curvature hypersurfaces [15], [22], [27], [30], [38], [62]. For the definition of (strong) stability for constant mean curvature hypersurfaces, see [8] and [30].…”

Section: Bernstein Problemmentioning

confidence: 99%

A Survey on Alexandrov-Bernstein-Hopf Theorems

Nelli¹

2008

View full text Add to dashboard Cite

show abstract

“…where B x 0 (R) denotes the geodesic ball of radius R centered at x 0 ∈ M. Many interesting generalizations of the do Carmo-Peng theorem have been obtained (see, e.g., [7,15,16,18]). By definition, the hyperbolic space H n+p is a Riemannian manifold with sectional curvature −1 which is simply connected, complete, and (n + p)-dimensional.…”

Section: Introductionmentioning

confidence: 99%

Rigidity of Complete Minimal Submanifolds in Spheres

Zhou

2020

Front. Phys.

View full text Add to dashboard Cite

Let M be an n-dimensional complete minimal submanifold in an (n + p)-dimensional sphere S n+p , and let h be the second fundamental form of M. In this paper, it is shown that M is totally geodesic if the L 2 norm of |h| on any geodesic ball of M is of less than quadratic growth and the L n norm of |h| on M is less than a fixed constant. Further, under only the latter condition, we prove that M is totally geodesic. Moreover, we provide a sufficient condition for a complete stable minimal hypersurface to be totally geodesic.

show abstract

Bernstein-type theorems in hypersurfaces with constant mean curvature

Abstract: By using the nodal domains of some natural function arising in the study of hypersurfaces with constant mean curvature we obtain some Bernstein-type theorems.

Cited by 12 publications

References 6 publications

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

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

A Survey on Alexandrov-Bernstein-Hopf Theorems

Rigidity of Complete Minimal Submanifolds in Spheres

Contact Info

Product

Resources

About