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Displaying similar documents to “Obata’s Rigidity Theorem for Metric Measure Spaces”

The Demyanov metric and some other metrics in the family of convex sets

Tadeusz Rzeżuchowski (2012)

Open Mathematics

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We describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.

Angles between Curves in Metric Measure Spaces

Bang-Xian Han, Andrea Mondino (2017)

Analysis and Geometry in Metric Spaces

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The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on RCD*(K, N) metric measure spaces. As a consequence,...

An area formula in metric spaces

Valentino Magnani (2011)

Colloquium Mathematicae

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We present an area formula for continuous mappings between metric spaces, under minimal regularity assumptions. In particular, we do not require any notion of differentiability. This is a consequence of a measure-theoretic notion of Jacobian, defined as the density of a suitable "pull-back measure". Finally, we give some applications and examples.

A note on global integrability of upper gradients of p-superharmonic functions

Outi Elina Maasalo, Anna Zatorska-Goldstein (2009)

Colloquium Mathematicae

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We consider a complete metric space equipped with a doubling measure and a weak Poincaré inequality. We prove that for all p-superharmonic functions there exists an upper gradient that is integrable on H-chain sets with a positive exponent.

Spaces of σ-finite linear measure

Ihor Stasyuk, Edward D. Tymchatyn (2013)

Colloquium Mathematicae

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Spaces of finite n-dimensional Hausdorff measure are an important generalization of n-dimensional polyhedra. Continua of finite linear measure (also called continua of finite length) were first characterized by Eilenberg in 1938. It is well-known that the property of having finite linear measure is not preserved under finite unions of closed sets. Mauldin proved that if X is a compact metric space which is the union of finitely many closed sets each of which admits a σ-finite linear...

Gluing Hyperconvex Metric Spaces

Benjamin Miesch (2015)

Analysis and Geometry in Metric Spaces

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We investigate how to glue hyperconvex (or injective) metric spaces such that the resulting space remains hyperconvex. We give two new criteria, saying that on the one hand gluing along strongly convex subsets and on the other hand gluing along externally hyperconvex subsets leads to hyperconvex spaces. Furthermore, we show by an example that these two cases where gluing works are opposed and cannot be combined.

The Besov capacity in metric spaces

Juho Nuutinen (2016)

Annales Polonici Mathematici

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We study a capacity theory based on a definition of Hajłasz-Besov functions. We prove several properties of this capacity in the general setting of a metric space equipped with a doubling measure. The main results of the paper are lower bound and upper bound estimates for the capacity in terms of a modified Netrusov-Hausdorff content. Important tools are γ-medians, for which we also prove a new version of a Poincaré type inequality.

Infinitesimal Structure of Differentiability Spaces, and Metric Differentiation

Jeff Cheeger, Bruce Kleiner, Andrea Schioppa (2016)

Analysis and Geometry in Metric Spaces

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We prove metric differentiation for differentiability spaces in the sense of Cheeger [10, 14, 27]. As corollarieswe give a new proof of one of the main results of [14], a proof that the Lip-lip constant of any Lip-lip space in the sense of Keith [27] is equal to 1, and new nonembeddability results.

A minorization of the first positive eigenvalue of the scalar laplacian on a compact Riemannian manifold

Jacek Komorowski

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CONTENTSIntroduction.......................................................................................................... 51. A parametrix of tho laplacian................................................................................ 72. An estimation of the differential of an eigenfunction of the laplacian......... 163. A normal chart on a neighbourhood of a geodesic........................................ 274. Minorization of the first positive eigenvalue of the laplacian............................

Strictly convex metric spaces with round balls and fixed points

Inese Bula (2005)

Banach Center Publications

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The paper introduces a notion of strictly convex metric space and strictly convex metric space with round balls. These objects generalize the well known concept of strictly convex Banach space. We prove some fixed point theorems in strictly convex metric spaces with round balls.