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

J. A. Gálvez; A. Martínez

Displaying similar documents to “Hypersurfaces with constant curvature in $ℝ^{n + 1}$ ”

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

Thomas Hasanis (1984)

Colloquium Mathematicae

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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...

Complete spacelike hypersurfaces with constant scalar curvature

Schi Chang Shu (2008)

Archivum Mathematicum

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In this paper, we characterize the $n$ -dimensional $(n \geq 3)$ complete spacelike hypersurfaces $M^{n}$ in a de Sitter space $S_{1}^{n + 1}$ with constant scalar curvature and with two distinct principal curvatures one of which is simple.We show that $M^{n}$ is a locus of moving $(n - 1)$ -dimensional submanifold $M_{1}^{n - 1} (s)$ , along $M_{1}^{n - 1} (s)$ the principal curvature $λ$ of multiplicity $n - 1$ is constant and $M_{1}^{n - 1} (s)$ is umbilical in $S_{1}^{n + 1}$ and is contained in an $(n - 1)$ -dimensional sphere $S^{n - 1} (c (s)) = E^{n} (s) \cap S_{1}^{n + 1}$ and is of constant curvature ${(\frac{d {log | λ^{2} - (1 - R) |^{1 / n}}}{d s})}^{2} - λ^{2} + 1$ ,where $s$ is the arc length of an orthogonal trajectory...

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 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.

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.

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

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

Cours de l'institut Fourier

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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.

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 $E_{1}^{4}$ .

Some aspects of the variational nature of mean curvature flow

Giovanni Bellettini, Luca Mugnai (2008)

Journal of the European Mathematical Society

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We show that the classical solution of the heat equation can be seen as the minimizer of a suitable functional defined in space-time. Using similar ideas, we introduce a functional $ℱ$ on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with $ℱ$ . We show some connections between minimizers of $ℱ$ and mean curvature flow.

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.

Curvature measures and fractals

Steffen Winter

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Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ’classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets $F \subseteq ℝ^{d}$ (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic...

Gauss curvature estimates for minimal graphs

Maria Nowak, Magdalena Wołoszkiewicz (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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We estimate the Gauss curvature of nonparametric minimal surfaces over the two-slit plane $ℂ ∖ ((- \infty, - 1] \cup [1, \infty))$ at points above the interval $(- 1, 1)$ .

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

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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A characterization of a certain real hypersurface of type $(A_{2})$ in a complex projective space

Byung Hak Kim, In-Bae Kim, Sadahiro Maeda (2017)

Czechoslovak Mathematical Journal

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In the class of real hypersurfaces $M^{2 n - 1}$ isometrically immersed into a nonflat complex space form ${\tilde{M}}_{n} (c)$ of constant holomorphic sectional curvature $c$ $(\neq 0)$ which is either a complex projective space $ℂ P^{n} (c)$ or a complex hyperbolic space $ℂ H^{n} (c)$ according as $c > 0$ or $c < 0$ , there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds....

Displaying similar documents to “Hypersurfaces with constant curvature in ℝ n + 1 ”

Displaying similar documents to “Hypersurfaces with constant curvature in $ℝ^{n + 1}$ ”