Complete spacelike hypersurfaces with constant scalar curvature

Schi Chang Shu

Displaying similar documents to “Complete spacelike hypersurfaces with constant scalar curvature”

Hypersurfaces with constant $k$ -th mean curvature in a Lorentzian space form

Shichang Shu (2010)

Archivum Mathematicum

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In this paper, we study $n (n \geq 3)$ -dimensional complete connected and oriented space-like hypersurfaces $M^{n}$ in an (n+1)-dimensional Lorentzian space form $M_{1}^{n + 1} (c)$ with non-zero constant $k$ -th $(k < n)$ mean curvature and two distinct principal curvatures $λ$ and $μ$ . We give some characterizations of Riemannian product $H^{m} (c_{1}) \times M^{n - m} (c_{2})$ and show that the Riemannian product $H^{m} (c_{1}) \times M^{n - m} (c_{2})$ is the only complete connected and oriented space-like hypersurface in $M_{1}^{n + 1} (c)$ with constant $k$ -th mean curvature and two distinct principal curvatures, if the multiplicities...

Hypersurfaces with constant curvature in $ℝ^{n + 1}$

J. A. Gálvez, A. Martínez (2002)

Banach Center Publications

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We give some optimal estimates of the height, curvature and volume of compact hypersurfaces in $ℝ^{n + 1}$ with constant curvature bounding a planar closed (n-1)-submanifold.

On convex hypersurfaces in $E^{n + 1}$

Thomas Hasanis (1984)

Colloquium Mathematicae

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New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino, Henrique F. de Lima (2015)

Archivum Mathematicum

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In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space $ℍ^{n + 1}$ , that is, complete hypersurfaces of $ℍ^{n + 1}$ whose mean curvature $H$ and normalized scalar curvature $R$ satisfy $R = a H + b$ for some $a$ , $b \in ℝ$ . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of $ℍ^{n + 1}$ . Furthermore,...

A characterization of n-dimensional hypersurfaces in $R^{n + 1}$ with commuting curvature operators

Yulian T. Tsankov (2005)

Banach Center Publications

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Let Mⁿ be a hypersurface in $R^{n + 1}$ . We prove that two classical Jacobi curvature operators $J_{x}$ and $J_{y}$ commute on Mⁿ, n > 2, for all orthonormal pairs (x,y) and for all points p ∈ M if and only if Mⁿ is a space of constant sectional curvature. Also we consider all hypersurfaces with n ≥ 4 satisfying the commutation relation $(K_{x, y} \circ K_{z, u}) (u) = (K_{z, u} \circ K_{x, y}) (u)$ , where $K_{x, y} (u) = R (x, y, u)$ , for all orthonormal tangent vectors x,y,z,w and for all points p ∈ M.

A new characterization of the sphere in $R^{3}$

Thomas Hasanis (1980)

Annales Polonici Mathematici

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Let M be a closed connected surface in $R^{3}$ with positive Gaussian curvature K and let $K_{I} I$ be the curvature of its second fundamental form. It is shown that M is a sphere if $K_{I} I = c \sqrt H K^{r}$ , for some constants c and r, where H is the mean curvature of M.

Sharp estimates for bubbling solutions of a fourth order mean field equation

Chang-Shou Lin, Juncheng Wei (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider a sequence of multi-bubble solutions $u_{k}$ of the following fourth order equation $Δ^{2} u_{k} = ρ_{k} \frac{h (x) e^{u_{k}}}{\int_{Ω} h e^{u_{k}}} in Ω, u_{k} = Δ u_{k} = 0 on \partial Ω, (*)$ where $h$ is a $C^{2, β}$ positive function, $Ω$ is a bounded and smooth domain in $ℝ^{4}$ , and $ρ_{k}$ is a constant such that $ρ_{k} \leq C$ . We show that (after extracting a subsequence), ${lim}_{k \to + \infty} ρ_{k} = 32 σ_{3} m$ for some positive integer $m \geq 1$ , where $σ_{3}$ is the area of the unit sphere in $ℝ^{4}$ . Furthermore, we obtain the following sharp estimates for $ρ_{k}$ : $\begin{matrix} ρ_{k} - 32 σ_{3} m & = c_{0} \sum_{j = 1}^{m} ϵ_{k, j}^{2} (\sum_{l \neq j} Δ G_{4} (p_{j}, p_{l}) + Δ R_{4} (p_{j}, p_{j}) + \frac{1}{32 σ_{3}} Δ log h (p_{j})) \\ + o (\sum_{j = 1}^{m} ϵ_{k, j}^{2}) \end{matrix}$ where $c_{0} > 0$ , $log \frac{64}{ϵ_{k, j}^{4}} = max_{x \in B_{δ} (p_{j})} u_{k} (x) - log (\int_{Ω} h e^{u_{k}})$ and $u_{k} \to 32 σ_{3} \sum_{j = 1}^{m} G_{4} (\cdot, p_{j})$ in $C_{loc}^{4} (Ω ∖ {p_{1}, ..., p_{m}})$ . This yields a bound of solutions as...

The resolution of the bounded $L^{2}$ curvature conjecture in general relativity

Sergiu Klainerman, Igor Rodnianski, Jérémie Szeftel (2014-2015)

Séminaire Laurent Schwartz — EDP et applications

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This paper reports on the recent proof of the bounded $L^{2}$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the $L^{2}$ -norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

Hypersurfaces with free boundary and large constant mean curvature: concentration along submanifolds

Mouhamed Moustapha Fall, Fethi Mahmoudi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given a domain $Ω$ of $ℝ^{m + 1}$ and a $k$ -dimensional non-degenerate minimal submanifold $K$ of $\partial Ω$ with $1 \leq k \leq m - 1$ , we prove the existence of a family of embedded constant mean curvature hypersurfaces in $Ω$ which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial Ω$ perpendicularly along their boundaries.

On Some Family of Contractible Hypersurfaces in $C^{4}$

S. Kaliman, L. Makar-Limanov (1993)

Cours de l'institut Fourier

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Mean curvature properties for $p$ -Laplace phase transitions

Berardino Sciunzi, Enrico Valdinoci (2005)

Journal of the European Mathematical Society

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This paper deals with phase transitions corresponding to an energy which is the sum of a kinetic part of $p$ -Laplacian type and a double well potential $h_{0}$ with suitable growth conditions. We prove that level sets of solutions of $Δ_{p} u = h_{0}^{'} (u)$ possessing a certain decay property satisfy a mean curvature equation in a suitable weak viscosity sense. From this, we show that, if the above level sets approach uniformly a hypersurface, the latter has zero mean curvature.

Tangency properties of sets with finite geometric curvature energies

Sebastian Scholtes (2012)

Fundamenta Mathematicae

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We investigate tangential regularity properties of sets of fractal dimension, whose inverse thickness or integral Menger curvature energies are bounded. For the most prominent of these energies, the integral Menger curvature $ℳ_{p}^{α} (X) : = \int_{X} \int_{X} \int_{X} κ^{p} (x, y, z) d_{X}^{α} (x) d_{X}^{α} (y) d_{X}^{α} (z)$ , where κ(x,y,z) is the inverse circumradius of the triangle defined by x,y and z, we find that $ℳ_{p}^{α} (X) < \infty$ for p ≥ 3α implies the existence of a weak approximate α-tangent at every point of the set, if some mild density properties hold. This includes the scale invariant...

Spacelike intersection curve of three spacelike hypersurfaces in $E_{1}^{4}$

B. Uyar Duldul, M. Caliskan (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space $E_{1}^{4}$ .

On some properties of a Riemannian space $V_{4}$ with constant scalar curvature

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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Global pinching theorems for minimal submanifolds in spheres

Kairen Cai (2003)

Colloquium Mathematicae

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Let M be a compact submanifold with parallel mean curvature vector embedded in the unit sphere $S^{n + p} (1)$ . By using the Sobolev inequalities of P. Li to get $L_{p}$ estimates for the norms of certain tensors related to the second fundamental form of M, we prove some rigidity theorems. Denote by H and ${| | σ | |}_{p}$ the mean curvature and the $L_{p}$ norm of the square length of the second fundamental form of M. We show that there is a constant C such that if ${| | σ | |}_{n / 2} < C$ , then M is a minimal submanifold in the sphere $S^{n + p - 1} (1 + H ²)$ with sectional...