Metric Measure Geometry: Gromov's Theory of Convergence and Concentration of Metrics and Measures

Metric Measure Geometry: Gromov's Theory of Convergence and Concentration of Metrics and Measures

Author
Takashi Shioya
Publisher
European Mathematical Society
Language
English
Year
2016
Page
196
ISBN
3037191589,9783037191583
File Type
pdf
File Size
1.1 MiB

This book studies a new theory of metric geometry on metric measure spaces. The theory was originally developed by M. Gromov in his book Metric Structures for Riemannian and Non-Riemannian Spaces and based on the idea of the concentration of measure phenomenon by Lévy and Milman. A central theme in this book is the study of the observable distance between metric measure spaces, defined by the difference between 1-Lipschitz functions on one space and those on the other. The topology on the set of metric measure spaces induced by the observable distance function is weaker than the measured Gromov Hausdorff topology and allows the author to investigate a sequence of Riemannian manifolds with unbounded dimensions. One of the main parts of this presentation is the discussion of a natural compactification of the completion of the space of metric measure spaces. The stability of the curvature-dimension condition is also discussed. A publication of the European Mathematical Society. Distributed within the Americas by the American Mathematical Society.

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