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

Thomas Hasanis

Displaying similar documents to “On convex hypersurfaces in $E^{n + 1}$ ”

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,...

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

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

Cours de l'institut Fourier

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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.

Weingarten hypersurfaces of the spherical type in Euclidean spaces

Cid D. F. Machado, Carlos M. C. Riveros (2020)

Commentationes Mathematicae Universitatis Carolinae

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We generalize a parametrization obtained by A. V. Corro in (2006) in the three-dimensional Euclidean space. Using this parametrization we study a class of oriented hypersurfaces $M^{n}$ , $n \geq 2$ , in Euclidean space satisfying a relation $\sum_{r = 1}^{n} {(- 1)}^{r + 1} r f^{r - 1} (\binom{n}{r}) H_{r} = 0,$ where $H_{r}$ is the $r$ th mean curvature and $f \in C^{\infty} (M^{n}; ℝ)$ , these hypersurfaces are called Weingarten hypersurfaces of the spherical type. This class of hypersurfaces includes the surfaces of the spherical type (Laguerré minimal surfaces). We characterize these hypersurfaces in terms...

A short note on $f$ -biharmonic hypersurfaces

Selcen Y. Perktaş, Bilal E. Acet, Adara M. Blaga (2020)

Commentationes Mathematicae Universitatis Carolinae

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In the present paper we give some properties of $f$ -biharmonic hypersurfaces in real space forms. By using the $f$ -biharmonic equation for a hypersurface of a Riemannian manifold, we characterize the $f$ -biharmonicity of constant mean curvature and totally umbilical hypersurfaces in a Riemannian manifold and, in particular, in a real space form. As an example, we consider $f$ -biharmonic vertical cylinders in $S^{2} \times ℝ$ .

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.

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.

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....

Lightlike hypersurfaces of an indefinite Kaehler manifold of a quasi-constant curvature

Dae Ho Jin, Jae Won Lee (2019)

Communications in Mathematics

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We study lightlike hypersurfaces $M$ of an indefinite Kaehler manifold $\bar{M}$ of quasi-constant curvature subject to the condition that the characteristic vector field $ζ$ of $\bar{M}$ is tangent to $M$ . First, we provide a new result for such a lightlike hypersurface. Next, we investigate such a lightlike hypersurface $M$ of $\bar{M}$ such that (1) the screen distribution $S (T M)$ is totally umbilical or (2) $M$ is screen conformal.

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 genericity theorem for algebraic stacks and essential dimension of hypersurfaces

Zinovy Reichstein, Angelo Vistoli (2013)

Journal of the European Mathematical Society

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We compute the essential dimension of the functors Forms $_{n, d}$ and Hypersurf $_{n, d}$ of equivalence classes of homogeneous polynomials in $n$ variables and hypersurfaces in $ℙ^{n - 1}$ , respectively, over any base field $k$ of characteristic $0$ . Here two polynomials (or hypersurfaces) over $K$ are considered equivalent if they are related by a linear change of coordinates with coefficients in $K$ . Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application...

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.

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 $E_{1}^{4}$ .

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.

$f$ -biminimal maps between Riemannian manifolds

Yan Zhao, Ximin Liu (2019)

Czechoslovak Mathematical Journal

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We give the definition of $f$ -biminimal submanifolds and derive the equation for $f$ -biminimal submanifolds. As an application, we give some examples of $f$ -biminimal manifolds. Finally, we consider $f$ -minimal hypersurfaces in the product space $ℝ^{n} \times 𝕊^{1} (a)$ and derive two rigidity theorems.

Hypersurfaces with singular locus a plane curve and transversal type $A_{1}$

Dirk Siersma (1988)

Banach Center Publications

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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 certain tensor on real hypersurfaces in a nonflat complex space form

Kazuhiro Okumura (2020)

Czechoslovak Mathematical Journal

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In a nonflat complex space form (namely, a complex projective space or a complex hyperbolic space), real hypersurfaces admit an almost contact metric structure $(φ, ξ, η, g)$ induced from the ambient space. As a matter of course, many geometers have investigated real hypersurfaces in a nonflat complex space form from the viewpoint of almost contact metric geometry. On the other hand, it is known that the tensor field $h$ $(= \frac{1}{2} ℒ_{ξ} φ)$ plays an important role in contact Riemannian geometry. In this...

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 $n = 1$ ....

Displaying similar documents to “On convex hypersurfaces in E n + 1 ”

Displaying similar documents to “On convex hypersurfaces in $E^{n + 1}$ ”