Curvature measures and fractals

Steffen Winter

Displaying similar documents to “Curvature measures and fractals”

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...

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.

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.

Functional inequalities and manifolds with nonnegative weighted Ricci curvature

Jing Mao (2020)

Czechoslovak Mathematical Journal

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We show that $n$ -dimensional $(n ⩾ 2)$ complete and noncompact metric measure spaces with nonnegative weighted Ricci curvature in which some Caffarelli-Kohn-Nirenberg type inequality holds are isometric to the model metric measure $n$ -space (i.e. the Euclidean metric $n$ -space). We also show that the Euclidean metric spaces are the only complete and noncompact metric measure spaces of nonnegative weighted Ricci curvature satisfying some prescribed Sobolev type inequality.

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.

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,...

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

W. Slósarska, Z. Żekanowski (1972)

Colloquium Mathematicae

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Geometry of Mus-Sasaki metric

Abderrahim Zagane, Mustapha Djaa (2018)

Communications in Mathematics

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In this paper, we introduce the Mus-Sasaki metric on the tangent bundle $T M$ as a new natural metric non-rigid on $T M$ . First we investigate the geometry of the Mus-Sasakian metrics and we characterize the sectional curvature and the scalar 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...

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 $(- 1, 1)$ .

$Q$ -curvature and spectral invariants

Branson, Thomas

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The Geometry of Differential Harnack Estimates

Sebastian Helmensdorfer, Peter Topping (2011-2012)

Séminaire de théorie spectrale et géométrie

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In this short note, we hope to give a rapid induction for non-experts into the world of Differential Harnack inequalities, which have been so influential in geometric analysis and probability theory over the past few decades. At the coarsest level, these are often mysterious-looking inequalities that hold for ‘positive’ solutions of some parabolic PDE, and can be verified quickly by grinding out a computation and applying a maximum principle. In this note we emphasise the geometry behind...

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.

On the total curvature of surfaces in $E^{4}$

Peter Wintgen (1978)

Colloquium Mathematicae

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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 $f : M^{2 n} \to ℝ^{2 n + p}$ 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 $ℝ^{3 n}$ 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...

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

Thomas Hasanis (1984)

Colloquium Mathematicae

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