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

Thomas Hasanis

Displaying similar documents to “A new characterization of the sphere in $R^{3}$ ”

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.

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...

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.

Two-dimensional curvature functionals with superquadratic growth

Ernst Kuwert, Tobias Lamm, Yuxiang Li (2015)

Journal of the European Mathematical Society

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For two-dimensional, immersed closed surfaces $f : Σ \to ℝ^{n}$ , we study the curvature functionals $ℰ^{p} (f)$ and $𝒲^{p} (f)$ with integrands ${(1 + | A |}^{2})^{p / 2}$ and ${(1 + | H |}^{2})^{p / 2}$ , respectively. Here $A$ is the second fundamental form, $H$ is the mean curvature and we assume $p > 2$ . Our main result asserts that $W^{2, p}$ critical points are smooth in both cases. We also prove a compactness theorem for $𝒲^{p}$ -bounded sequences. In the case of $ℰ^{p}$ this is just Langer’s theorem [16], while for $𝒲^{p}$ we have to impose a bound for the Willmore energy strictly below $8 π$ as an additional...

A geometric problem and the Hopf Lemma. I

Yan Yan Li, Louis Nirenberg (2006)

Journal of the European Mathematical Society

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A classical result of A. D. Alexandrov states that a connected compact smooth $n$ -dimensional manifold without boundary, embedded in $ℝ^{n + 1}$ , and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n + 1} = const$ in case $M$ satisfies: for any two points $(X^{'}, X_{n + 1})$ , $(X^{'}, {\hat{X}}_{n + 1})$ on $M$ , with $X_{n + 1} > {\hat{X}}_{n + 1}$ , the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n = 1$ ....

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

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Let $(M, g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$ , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$ . In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group $O (4)$ acts as a transformation group between ST bases at $T_{p} M$ and for the so-called 2-stein curvature tensors, the group $Sp (1) \subset SO (4)$ acts as a transformation...

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.

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

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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Travelling graphs for the forced mean curvature motion in an arbitrary space dimension

Régis Monneau, Jean-Michel Roquejoffre, Violaine Roussier-Michon (2013)

Annales scientifiques de l'École Normale Supérieure

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We construct travelling wave graphs of the form $z = - c t + φ (x)$ , $φ : x \in ℝ^{N - 1} \mapsto φ (x) \in ℝ$ , $N \geq 2$ , solutions to the $N$ -dimensional forced mean curvature motion $V_{n} = - c_{0} + κ$ ( $c \geq c_{0}$ ) with prescribed asymptotics. For any $1$ -homogeneous function $φ_{\infty}$ , viscosity solution to the eikonal equation $| D φ_{\infty} | = \sqrt{{(c / c_{0})}^{2} - 1}$ , we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by $φ_{\infty}$ . We also describe $φ_{\infty}$ in terms of a probability measure on $§^{N - 2}$ .

Upper bounds on the length of a shortest closed geodesic and quantitative Hurewicz theorem

Alexander Nabutovsky, Regina Rotman (2003)

Journal of the European Mathematical Society

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In this paper we present two upper bounds on the length of a shortest closed geodesic on compact Riemannian manifolds. The first upper bound depends on an upper bound on sectional curvature and an upper bound on the volume of the manifold. The second upper bound will be given in terms of a lower bound on sectional curvature, an upper bound on the diameter and a lower bound on the volume. The related questions that will also be studied are the following: given a contractible $k$ -dimensional...

A strong maximum principle for the Paneitz operator and a non-local flow for the $Q$ -curvature

Matthew J. Gursky, Andrea Malchiodi (2015)

Journal of the European Mathematical Society

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In this paper we consider Riemannian manifolds $(M^{n}, g)$ of dimension $n \geq 5$ , with semi-positive $Q$ -curvature and non-negative scalar curvature. Under these assumptions we prove (i) the Paneitz operator satisfies a strong maximum principle; (ii) the Paneitz operator is a positive operator; and (iii) its Green’s function is strictly positive. We then introduce a non-local flow whose stationary points are metrics of constant positive $Q$ -curvature. Modifying the test function construction of Esposito-Robert,...

A Study on $φ$ -recurrence $τ$ -curvature tensor in $(k, μ)$ -contact metric manifolds

Gurupadavva Ingalahalli, C.S. Bagewadi (2018)

Communications in Mathematics

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In this paper we study $φ$ -recurrence $τ$ -curvature tensor in $(k, μ)$ -contact metric manifolds.

Regularity of stable solutions of $p$ -Laplace equations through geometric Sobolev type inequalities

Daniele Castorina, Manel Sanchón (2015)

Journal of the European Mathematical Society

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We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $- Δ_{p} u = g (u)$ in a smooth bounded domain $Ω \subset ℝ^{n}$ . In particular, we obtain new $L^{r}$ and $W^{1, r}$ bounds for the extremal solution $u^{☆}$ when the domain is strictly convex. More precisely, we prove that $u^{☆} \in L^{\infty} (Ω)$ if $n \leq p + 2$ and $u^{☆} \in L^{\frac{n p}{n - p - 2}} (Ω) \cap W_{0}^{1, p} (Ω)$ if $n > p + 2$ .

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 $(- 1, 1)$ .

A half-space type property in the Euclidean sphere

Marco Antonio Lázaro Velásquez (2022)

Archivum Mathematicum

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We study the notion of strong $r$ -stability for the context of closed hypersurfaces $Σ^{n}$ ( $n \geq 3$ ) with constant $(r + 1)$ -th mean curvature $H_{r + 1}$ immersed into the Euclidean sphere $𝕊^{n + 1}$ , where $r \in {1, ..., n - 2}$ . In this setting, under a suitable restriction on the $r$ -th mean curvature $H_{r}$ , we establish that there are no $r$ -strongly stable closed hypersurfaces immersed in a certain region of $𝕊^{n + 1}$ , a region that is determined by a totally umbilical sphere of $𝕊^{n + 1}$ . We also provide a rigidity result for such hypersurfaces.

Structure of second-order symmetric Lorentzian manifolds

Oihane F. Blanco, Miguel Sánchez, José M. Senovilla (2013)

Journal of the European Mathematical Society

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$𝑆𝑒𝑐𝑜𝑛𝑑 - 𝑜𝑟𝑑𝑒𝑟𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐𝐿𝑜𝑟𝑒𝑛𝑡𝑧𝑖𝑎𝑛𝑠𝑝𝑎𝑐𝑒𝑠$ , that is to say, Lorentzian manifolds with vanishing second derivative $\nabla \nabla R \equiv 0$ of the curvature tensor $R$ , are characterized by several geometric properties, and explicitly presented. Locally, they are a product $M = M_{1} \times M_{2}$ where each factor is uniquely determined as follows: $M_{2}$ is a Riemannian symmetric space and $M_{1}$ is either a constant-curvature Lorentzian space or a definite type of plane wave generalizing the Cahen–Wallach family. In the proper case (i.e., $\nabla R \neq 0$ at some point), the 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...

$Γ$ -convergence of discrete approximations to interfaces with prescribed mean curvature

Giovanni Bellettini, Maurizio Paolini, Claudio Verdi (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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The numerical approximation of the minimum problem: ${min}_{A \subseteq Ω} \tilde{F} (A)$ , is considered, where $\tilde{F} (A) = P_{Ω} (A) + \cos (θ) H^{n - 1} (\partial A \cap \partial Ω) - \int_{A} κ$ . The solution to this problem is a set $A \subseteq Ω \subset R^{n}$ with prescribed mean curvature $κ$ and contact angle $θ$ at the intersection of $\partial A$ with $\partial Ω$ . The functional $\tilde{F}$ is first relaxed with a sequence of nonconvex functionals defined in $H^{1} (Ω)$ which, in turn, are discretized by finite elements. The $Γ$ -convergence of the discrete functionals to $\tilde{F}$ as well as the compactness of any sequence of discrete absolute minimizers are proven. ...

Displaying similar documents to “A new characterization of the sphere in R 3 ”

Displaying similar documents to “A new characterization of the sphere in $R^{3}$ ”