Two-ended hypersurfaces with zero scalar curvature

Hounie, Jorge; Leite, Maria Luiza

doi:10.1512/iumj.1999.48.1664

Cited by 35 publications

(34 citation statements)

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“…It was proved by Hounie and Leite in a general point of view, see [9]. In fact, P 1 positive definite implies L 1 (f ) = div(P 1 (∇f )) is an elliptic differential operator.…”

Section: Preliminary Resultsmentioning

confidence: 94%

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On stable hypersurfaces with vanishing scalar curvature

Neto

2013

Math. Z.

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We will prove that there are no stable complete hypersurfaces of R 4 with zero scalar curvature, polynomial volume growth and such that (−K) H 3 ≥ c > 0 everywhere, for some constant c > 0, where K denotes the Gauss-Kronecker curvature and H denotes the mean curvature of the immersion. Our second result is the Bernstein type one there is no entire graphs of R 4 with zero scalar curvature such that (−K) H 3 ≥ c > 0 everywhere. At last, it will be proved that, if there exists a stable hypersurface with zero scalar curvature and (−K) H 3 ≥ c > 0 everywhere, that is, with volume growth greater than polynomial, then its tubular neighborhood is not embedded for suitable radius.

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“…It was proved by Hounie and Leite in a general point of view, see [9]. In fact, P 1 positive definite implies L 1 (f ) = div(P 1 (∇f )) is an elliptic differential operator.…”

Section: Preliminary Resultsmentioning

confidence: 94%

“…In [9], Hounie and Leite showed that L 1 is elliptic if and only if rank A > 1. Thus, K = 0 everywhere implies L 1 is elliptic, and if H > 0, then P 1 is a positive definite linear operator.…”

Section: Introductionmentioning

confidence: 99%

On stable hypersurfaces with vanishing scalar curvature

Neto

2013

Math. Z.

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“…Observe that there are many examples of zero scalar curvature hyper-surfaces with finite total curvature which are not flat. See the examples provided by Lounie and Leite [13]. Clearly the analogy of our previous theorem with only the scalar flat assumption cannot be true.…”

Section: Theorem 11 Let M N (N > 2) Be a Complete And Noncompact Hypmentioning

confidence: 92%

“…Hence, it will be equally interesting to consider other elementary symmetric functions of the second fundamental form. In particular, it is natural to ask whether hyper-surfaces with zero scalar curvature have Bernstein type property [2,6,10,17]. However, observe that the equation for the hyper-surfaces with zero mean curvature is elliptic, the analogy for surfaces with zero scalar curvature is only a degenerate elliptic equation.…”

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confidence: 99%

The complete hyper-surfaces with zero scalar curvature in $$\mathbb{R }^{n+1}$$

Yaowen

Zhou

2013

Ann Glob Anal Geom

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Let M n be a complete and noncompact hyper-surface immersed in R n+1 . We should show that if M is of finite total curvature and Ricci flat, then M turns out to be a hyperplane. Meanwhile, the hyper-surfaces with the vanishing scalar curvature is also considered in this paper. It can be shown that if the total curvature is sufficiently small, then by refined Kato's inequality, conformal flatness and flatness are equivalent in some sense. And those results should be compared with Hartman and Nirenberg's similar results with flat curvature assumption.

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“…. , k n of an orientable hypersurface x : M n → R n+1 given by A breakthrough in the study of these hypersurfaces occurred in the last five years of last century: in (Hounie and Leite 1995) and (Hounie and Leite 1999) conditions for the linearization of the partial differential equation S r+1 = 0 to be an elliptic equation were found. This linearization involves a second order differential operator L r (see the definition of L r in Section 2) and the Hounie-Leite conditions read as follows:…”

Section: Introductionmentioning

confidence: 99%