Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

Bérard, Pierre; Carmo, Manfredo do; Santos, Walcy

doi:10.1007/978-3-642-25588-5_26

Cited by 17 publications

(20 citation statements)

References 15 publications

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“…Following the general ideas of [24], we use (4.36) and (4.37), to estimate the L p -norms of u and the classical de Giorgi-Moser-Nash method to estimate u ∞ outside big balls. The details appear in the proof of Theorem 4.1, p. 282 of [4], where it is observed that the proof only uses the facts that u satisfies Simons' inequality and M a Sobolev inequality. Theorem, Assertion 2.…”

Section: Translation Invariant Minimal Hypersurfaces In H N × Rmentioning

confidence: 99%

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Minimal hypersurfaces in $$\mathbb {H}^n \times \mathbb {R}$$ H n × R , total curvature and index

Bérard

Earp

2016

Boll Unione Mat Ital

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In this paper, we consider minimal hypersurfaces in the product space H n × R. We begin by studying examples of rotation hypersurfaces and hypersurfaces invariant under hyperbolic translations. We then consider minimal hypersurfaces with finite total curvature. This assumption implies that the corresponding curvature goes to zero uniformly at infinity. We show that surfaces with finite total intrinsic curvature have finite index. The converse statement is not true as shown by our examples which also serve as useful barriers.MSC (2000): 53C42, 58C40.

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Section: Translation Invariant Minimal Hypersurfaces In H N × Rmentioning

confidence: 99%

“…The de Giorgi-Moser-Nash technique applies (see [4], Theorem 4.1) and it follows that |A M | tends to zero uniformly at infinity.…”

Section: Translation Invariant Minimal Hypersurfaces In H N × Rmentioning

confidence: 99%

Minimal hypersurfaces in $$\mathbb {H}^n \times \mathbb {R}$$ H n × R , total curvature and index

Bérard

Earp

2016

Boll Unione Mat Ital

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“…Zhu stated that the previous result holds for any n but in the proof, they use an unpublished result by Anderson. P. Bérard, M. do Carmo and W. Santos [11] proved that, if M is a complete hypersurface in H n+1 , with constant mean curvature H, H 2 < 1, such that M |A − HI| n < ∞, then M has finite index. Notice that the converse is not true, as it is showed by the examples by A. da Silveira [19].…”

Section: Stability Notions and Finite Index Hypersurfacesmentioning

confidence: 99%

“…where R ′w and R(A) w are defined as follows: In order to compute the first term n i,j=1 ∇ 2 B(e i , e j ), B(e i , e j ) in (10), we take x = e i , y = e j , w = B(e i , e j ) in (11) and sum on i and j. The computation of all the terms is as follows.…”

Section: Simons' Inequality For Constant Mean Curvature Hypersurfacesmentioning

confidence: 99%

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

Ilias¹,

Nelli²,

Soret³

2012

Ann Glob Anal Geom

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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.

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“…In particular, |Φ| ≡ 0 if and only if the immersion x is totally umbilical. We say that the immersion x has finite total curvature if the L m -norm of the traceless second fundamental form is finite (see [4] and references therein), that is,…”

Section: Introductionmentioning

confidence: 99%

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

2012

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Let x : M m →M , m ≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply-connected manifoldM with sectional curvature satisfying −k 2 ≤ KM ≤ 0, for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L m -norm. If KM ≡ 0, assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L 2 harmonic 1-forms on M has finite dimension. Moreover there exists a constant Λ > 0, explicitly computed, such that if the total curvature is bounded from above by Λ then there is no nontrivial L 2 -harmonic 1-forms on M .

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Complete Hypersurfaces with Constant Mean Curvature and Finite Total Curvature

Cited by 17 publications

References 15 publications

Minimal hypersurfaces in $$\mathbb {H}^n \times \mathbb {R}$$ H n × R , total curvature and index

Minimal hypersurfaces in $$\mathbb {H}^n \times \mathbb {R}$$ H n × R , total curvature and index

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

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

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