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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 Proof of a General Isoperimetric Inequality for Surfaces.

Manfredo do Carmo, Joao Lucas Barbosa (1978)

Mathematische Zeitschrift

A typical convex surface contains no closed geodesic.

Peter M. Gruber (1991)

Journal für die reine und angewandte Mathematik

A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature

Vladimir Oliker (2005)

Banach Center Publications

In his book on convex polytopes [2] A. D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in $ℝ^{n + 1}$ , n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature problem [20]. In this paper we give a simple variational proof of existence for the A. D. Aleksandrov problem...

Adaptation globale de la méthode de Weingarten et réalisations de disques ajustés

Philippe Delanoë (1987/1988)

Séminaire de théorie spectrale et géométrie

Alexandrov Par Excellence.

Kutateladze, S.S. (2007)

Sibirskij Matematicheskij Zhurnal

Alexandrov spaces with nonnegative curvature outside a compact set.

Liang-Khoon Koh (1997)

Manuscripta mathematica

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

Alexander I. Bobenko, Ivan Izmestiev (2008)

Annales de l’institut Fourier

We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of generalized convex...

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Displaying 1 – 20 of 159

A counterexample to the rigidity conjecture for polyhedra

A. D. Alexandrov's uniqueness theorem for convex polytopes and its refinements.

A direct method approach for the existence of convex hypersurfaces with prescribed Gauss-Kronecker curvature.

A Helmholtz-Lie Type Characterization of Ellipsoids, I .

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

A Proof of a General Isoperimetric Inequality for Surfaces.

A typical convex surface contains no closed geodesic.

A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature

Adaptation globale de la méthode de Weingarten et réalisations de disques ajustés

Alexandrov Par Excellence.

Alexandrov spaces with nonnegative curvature outside a compact set.

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

An integral formula for closed surfaces and a generalization of $H p$ -theorem

$C^{1}$ -гладкие изометрические погружения

$C^{1}$ -гладкие поверхности ограниченной внешней положительной кривизны

Characterizations of the sphere by the curvature of the second fundamental form

Collapsing and soul theorem in three-dimension

Convex functionals and generalized harmonic maps into spaces of non positive curvature.

Convex Surfaces Whose Geodesics Are Planar.

Corrections to my paper "On the geometry of convex reflectors" (Banach Center Publ. 57 (2002), 155-169)

Currently displaying 1 – 20 of 159