A topological splitting theorem for weighted Alexandrov spaces

Kuwae, Kazuhiro; Shioya, Takashi

doi:10.2748/tmj/1303219936

Cited by 19 publications

(22 citation statements)

References 38 publications

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“…The proof is similar to [KS,Lemma 5.6] (see also [EH], [FLZ,Lemma 2.1]) thanks to the Laplacian comparison theorem (Theorem 2.8).…”

Section: Analysis Of Busemann Functionsmentioning

confidence: 72%

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Ohta

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We investigate the structure of a Finsler manifold of nonnegative weighted Ricci curvature including a straight line, and extend the classical Cheeger-GromollLichnerowicz splitting theorem. Such a space admits a diffeomorphic, measurepreserving splitting in general. As for a special class of Berwald spaces, we can perform the isometric splitting in the sense that there is a one-parameter family of isometries generated from the gradient vector field of the Busemann function. A Betti number estimate is also given for Berwald spaces.

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“…The proof is similar to [KS,Lemma 5.6] (see also [EH], [FLZ,Lemma 2.1]) thanks to the Laplacian comparison theorem (Theorem 2.8).…”

Section: Analysis Of Busemann Functionsmentioning

confidence: 72%

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Ohta

2013

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Kuwae and Shioya [KS3] consider (weighted) Alexandrov spaces of curvature ≥ −1 with nonnegative Ricci curvature in terms of the measure contraction property (see Subsection 8.3). They show that, if such an Alexandrov space contains an isometric copy of the real line, then it splits off R as topological measure spaces (compare this with Theorem 6.6).…”

Section: Open Questionsmentioning

confidence: 99%

Ricci curvature, entropy, and optimal transport

Ohta

2014

Optimal Transport

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This is the lecture notes on the interplay between optimal transport and Riemannian geometry. On a Riemannian manifold, the convexity of entropy along optimal transport in the space of probability measures characterizes lower bounds of the Ricci curvature. We then discuss geometric properties of general metric measure spaces satisfying this convexity condition. Mathematics Subject Classification (2000): 53C21, 53C23, 53C60, 28A33, 28D20Keywords: Ricci curvature, entropy, optimal transport, curvature-dimension condition IntroductionThis article is extended notes based on the author's lecture series in summer school at Université Joseph Fourier, Grenoble: 'Optimal Transportation: Theory and Applications'. The aim of these five lectures (corresponding to Sections 3-7) was to review the recent impressive development on the interplay between optimal transport theory and Riemannian geometry. Ricci curvature and entropy are the key ingredients. See [Lo2] for a survey in the same spirit with a slightly different selection of topics.Optimal transport theory is concerned with the behavior of transport between two probability measures in a metric space. We say that such transport is optimal if it minimizes a certain cost function typically defined from the distance of the metric space. Optimal transport naturally inherits the geometric structure of the underlying space, especially Ricci curvature plays a crucial role for describing optimal transport in Riemannian manifolds. In fact, optimal transport is always performed along geodesics, and we obtain Jacobi fields as their variational vector fields. The behavior of these Jacobi fields is controlled by the Ricci curvature as is usual in comparison geometry. In this way, a lower Ricci curvature bound turns out to be equivalent to a certain convexity property of entropy in the space of probability measures. The latter convexity condition is called the curvature-dimension condition, and it can be formulated without using the differentiable * Partly supported by the Grant-in-Aid for Young Scientists (B) 20740036. 1 structure. Therefore the curvature-dimension condition can be regarded as a 'definition' of a lower Ricci curvature bound for general metric measure spaces, and implies many analogous properties in an interesting way.A prerequisite is the basic knowledge of optimal transport theory and Wasserstein geometry. Riemannian geometry is also necessary in Sections 3, 4, and is helpful for better understanding of the other sections. We refer to [AGS] The organization of this article is as follows. After summarizing some notations we use, Section 3 is devoted to the definition of the Ricci curvature of Riemannian manifolds and to the classical Bishop-Gromov volume comparison theorem. In Section 4, we start with the Brunn-Minkowski inequalities in (unweighted or weighted) Euclidean spaces, and explain the equivalence between a lower (weighted) Ricci curvature bound for a (weighted) Riemannian manifold and the curvature-dimension condition. In Section 5, we give the prec...

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“…on Ω. Now we can define For which we have the following maximum principe of Kuwae-Shioya [26] Theorem 4.2. Let f ∈ W 1,2 0,loc (Ω; µ) be a continuous µ-subharmonic function and suppose f attains its maximum on Ω.…”

Section: Harmonic Functionsmentioning

confidence: 99%

Alexandrov spaces with large volume growth

Munn

2014

Journal of Mathematical Analysis and Applications

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Let (X, d) be an n-dimensional Alexandrov space whose Hausdorff measure H n satisfies a condition giving the metric measure space (X, d, H n ) a notion of having nonnegative Ricci curvature. We examine the influence of large volume growth on these spaces and generalize some classical arguments from Riemannian geometry showing that when the volume growth is sufficiently large, then (X, d, H n ) has finite topological type.

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A topological splitting theorem for weighted Alexandrov spaces

Cited by 19 publications

References 38 publications

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Splitting theorems for Finsler manifolds of nonnegative Ricci curvature

Ricci curvature, entropy, and optimal transport

Alexandrov spaces with large volume growth

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