A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces

A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces Potential Anal https://doi.org/10.1007/s11118-018-9708-4 A Sufficient Condition to a Regular Set Being of Positive Measure on RCD Spaces Yu Kitabeppu Received: 12 February 2018 / Accepted: 14 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract In this paper, we study regular sets in metric measure spaces with Ricci curvature bounded from below. We prove that the existence of a point in the regular set of the highest dimension implies the positivity of the measure of such regular set. Also we define the dimension of RCD spaces and prove the lower semicontinuity of that under the Gromov- Hausdorff convergence. Keywords RCD spaces · Regular sets Mathematics Subject Classification (2010) Primary 51F99 · Secondary 53C20 1 Introduction In the series of papers [13–15] by Cheeger and Colding, they investigate much properties of Ricci limit spaces. Especially, they show that the study of the infinitesimal structure on such spaces is pretty important to understand the geometry of that. On a non-collapsing Ricci limit space (Y,d,ν), ν-almost every point has the unique tangent cone that is isometric to N-dimensional Euclidean space when the sequence of Rimannian manifolds approximating Y are of N-dimension [13]. It is also known that the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Potential Analysis Springer Journals

A Sufficient Condition to a Regular Set Being of Positive Measure on Spaces

Potential Analysis , Volume OnlineFirst – May 29, 2018
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Publisher
Springer Netherlands
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Mathematics; Potential Theory; Probability Theory and Stochastic Processes; Geometry; Functional Analysis
ISSN
0926-2601
eISSN
1572-929X
D.O.I.
10.1007/s11118-018-9708-4
Publisher site
See Article on Publisher Site

Abstract

Potential Anal https://doi.org/10.1007/s11118-018-9708-4 A Sufficient Condition to a Regular Set Being of Positive Measure on RCD Spaces Yu Kitabeppu Received: 12 February 2018 / Accepted: 14 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract In this paper, we study regular sets in metric measure spaces with Ricci curvature bounded from below. We prove that the existence of a point in the regular set of the highest dimension implies the positivity of the measure of such regular set. Also we define the dimension of RCD spaces and prove the lower semicontinuity of that under the Gromov- Hausdorff convergence. Keywords RCD spaces · Regular sets Mathematics Subject Classification (2010) Primary 51F99 · Secondary 53C20 1 Introduction In the series of papers [13–15] by Cheeger and Colding, they investigate much properties of Ricci limit spaces. Especially, they show that the study of the infinitesimal structure on such spaces is pretty important to understand the geometry of that. On a non-collapsing Ricci limit space (Y,d,ν), ν-almost every point has the unique tangent cone that is isometric to N-dimensional Euclidean space when the sequence of Rimannian manifolds approximating Y are of N-dimension [13]. It is also known that the

Journal

Potential AnalysisSpringer Journals

Published: May 29, 2018

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