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An intrinsic rigidity theorem for closed minimal hypersurfaces in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature  [PDF]
Bing Tang,Ling Yang
Mathematics , 2015,
Abstract: Let M be a closed minimal hypersurface in 5-dimensional Euclidean sphere with constant nonnegative scalar curvature. We prove that, if the sum of the cubes of all principal curvatures and the number of distinct principal curvatures are constant, then M is isoparametric. Moreover, We give all possible values for squared length of the second fundamental form of M. This result provides another piece of supporting evidence to the Chern conjecture.
Stable hypersurfaces with zero scalar curvature in Euclidean space  [PDF]
Hilário Alencar,Manfredo do Carmo,Gregório Silva Neto
Mathematics , 2015,
Abstract: In this paper we prove some results concerning stability of hypersurfaces in the four dimensional Euclidean space with zero scalar curvature. First we prove there is no complete stable hypersurface with zero scalar curvature, polynomial growth of integral of the mean curvature, and with the Gauss-Kronecker curvature bounded away from zero. We conclude this paper giving a sufficient condition for a regular domain to be stable in terms of the mean and the Gauss-Kronecker curvatures of the hypersurface and the extrinsic diameter of the domain.
Translation hypersurfaces with constant Sr curvature in the Euclidean space  [PDF]
Barnabe Pessoa Lima,Paulo Alexandre Araujo Sousa,Juscelino Pereira Silva,Newton Luis Santos
Mathematics , 2014,
Abstract: In this paper, we give a complete description of all translation hypersurfaces with constant r-curvature Sr, in the Euclidean space.
On stable hypersurfaces with constant mean curvature in Euclidean spaces  [PDF]
Jinpeng Lu
Mathematics , 2012,
Abstract: In this paper, we derive curvature estimates for strongly stable hypersurfaces with constant mean curvature immersed in $\mathbb{R}^{n+1}$, which show that the locally controlled volume growth yields a globally controlled volume growth if $\partial M=\emptyset$. Moreover, we deduce a Bernstein-type theorem for complete stable hypersurfaces with constant mean curvature of arbitrary dimension, given a finite $L^p$-norm curvature condition.
Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space  [PDF]
Pierre Bayard
Mathematics , 2008,
Abstract: We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates.
Isoparametric hypersurfaces and metrics of constant scalar curvature  [PDF]
Guillermo Henry,Jimmy Petean
Mathematics , 2011,
Abstract: We showed the existence of non-radial solutions of the equation $\Delta u -\lambda u + \lambda u^q =0$ on the round sphere $S^m$, for $q<2m/(m-2)$, and study the number of such solutions in terms of $\lambda$. We show that for any isoparametric hypersurface $M\subset S^m$ there are solutions such that $M$ is a regular level set (and the number of such solutions increases with $\lambda$). We also show similar results for isoparametric hypersurfaces in general Riemannian manifolds. These solutions give multiplicity results for metrics of constant scalar curvature on conformal classes of Riemannian products.
Constant scalar curvature hypersurfaces in extended Schwarzschild space-time  [PDF]
M. J. Pareja,J. Frauendiener
Physics , 2006, DOI: 10.1103/PhysRevD.74.044026
Abstract: We present a class of spherically symmetric hypersurfaces in the Kruskal extension of the Schwarzschild space-time. The hypersurfaces have constant negative scalar curvature, so they are hyperboloidal in the regions of space-time which are asymptotically flat.
Some geometric properties of hypersurfaces with constant $r$-mean curvature in Euclidean space  [PDF]
Debora Impera,Luciano Mari,Marco Rigoli
Mathematics , 2010,
Abstract: Let $f:M\ra \erre^{m+1}$ be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors in \cite{bimari} to analyze the stability of the differential operator $L_r$ associated with the $r$-th Newton tensor of $f$. This appears in the Jacobi operator for the variational problem of minimizing the $r$-mean curvature $H_r$. Two natural applications are found. The first one ensures that, under the mild condition that the integral of $H_r$ over geodesic spheres grows sufficiently fast, the Gauss map meets each equator of $\esse^m$ infinitely many times. The second one deals with hypersurfaces with zero $(r+1)$-mean curvature. Under similar growth assumptions, we prove that the affine tangent spaces $f_*T_pM$, $p\in M$, fill the whole $\erre^{m+1}$.
On the scalar curvature of constant mean curvature hypersurfaces in space forms  [PDF]
Luis J. Alias,S. Carolina Garcia-Martinez
Mathematics , 2009,
Abstract: In this paper we study the behavior of the scalar curvature $S$ of a complete hypersurface immersed with constant mean curvature into a Riemannian space form of constant curvature, deriving a sharp estimate for the infimum of $S$. Our results will be an application of a weak Omori-Yau maximum principle due to Pigola, Rigoli and Setti \cite{PRS}.
Second eigenvalue of a Jacobi operator of hypersurfaces with constant scalar curvature  [PDF]
Haizhong Li,Xianfeng Wang
Mathematics , 2010,
Abstract: Let $x:M\to\mathbb{S}^{n+1}(1)$ be an n-dimensional compact hypersurface with constant scalar curvature $n(n-1)r,~r\geq 1$, in a unit sphere $\mathbb{S}^{n+1}(1),~n\geq 5$. We know that such hypersurfaces can be characterized as critical points for a variational problem of the integral $\int_MH dv$ of the mean curvature $H$. In this paper, we derive an optimal upper bound for the second eigenvalue of the Jacobi operator $J_s$ of $M$. Moreover, when $r>1$, the bound is attained if and only if $M$ is totally umbilical and non-totally geodesic, when $r=1$, the bound is attained if $M$ is the Riemannian product $\mathbb{S}^{m}(c)\times\mathbb{S}^{n-m}(\sqrt{1-c^2}),~1\leq m\leq n-2,~c=\sqrt{\frac{(n-1)m+\sqrt{(n-1)m(n-m)}}{n(n-1)}}$.
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