Displaying similar documents to “On the total curvature of surfaces in E 4

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 d (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic...

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 p α ( X ) : = X X 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 p α ( X ) < 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...

On the motion of a curve by its binormal curvature

Jerrard, Robert L., Didier Smets (2015)

Journal of the European Mathematical Society

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We propose a weak formulation for the binormal curvature flow of curves in 3 . This formulation is sufficiently broad to consider integral currents as initial data, and sufficiently strong for the weak-strong uniqueness property to hold, as long as self-intersections do not occur. We also prove a global existence theorem in that framework.

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 H K r , for some constants c and r, where H is the mean curvature of M.

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.

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 ( ( - , - 1 ] [ 1 , ) ) at points above the interval ( - 1 , 1 ) .

On real Kähler Euclidean submanifolds with non-negative Ricci curvature

Luis A. Florit, Wing San Hui, F. Zheng (2005)

Journal of the European Mathematical Society

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We show that any real Kähler Euclidean submanifold f : M 2 n 2 n + p with either non-negative Ricci curvature or non-negative holomorphic sectional curvature has index of relative nullity greater than or equal to 2 n 2 p . Moreover, if equality holds everywhere, then the submanifold must be a product of Euclidean hypersurfaces almost everywhere, and the splitting is global provided that M 2 n is complete. In particular, we conclude that the only real Kähler submanifolds M 2 n in 3 n that have either positive Ricci curvature...

On Uniqueness Theoremsfor Ricci Tensor

Marina B. Khripunova, Sergey E. Stepanov, Irina I. Tsyganok, Josef Mikeš (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold M and a symmetric 2-tensor r , construct a metric on M whose Ricci tensor equals r . In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita connection on any compact Einstein manifold with non-negative section curvature. In the present paper we generalize the result of DeTurck and Koiso for a Riemannian manifold with...

Hypersurfaces with constant curvature in 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 n + 1 with constant curvature bounding a planar closed (n-1)-submanifold.

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 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 ) × M n - m ( c 2 ) and show that the Riemannian product H m ( c 1 ) × 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...

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 on the class of space-time tracks of moving hypersurfaces, and we study suitable minimization problems related with . We show some connections between minimizers of and mean curvature flow.