Lecture Notes in Mathematics
© 2020
Geometry and Analysis of Metric Spaces via Weighted Partitions
Authors: Kigami, Jun
Free Preview- Describes how a compact metric space may be associated with an infinite graph whose boundary is the original space
- Explores an approach to metrics and measures from an integrated point of view
- Shows a relation between geometry (Ahlfors regular conformal dimension) and analysis (critical index of p-energies)
Buy this book
- About this book
-
The aim of these lecture notes is to propose a systematic framework for geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space. Metrics and measures on the space are then studied from an integrated point of view as weights of the partition. In the course of the text:
- It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic.
- Various relations between metrics and measures such as bilipschitz equivalence, quasisymmetry, Ahlfors regularity, and the volume doubling property are translated to relations between weights. In particular, it is shown that the volume doubling property between a metric and a measure corresponds to a quasisymmetry between two metrics in the language of weights.
- The Ahlfors regular conformal dimension of a compact metric space is characterized as the critical index of p-energies associated with the partition and the weight function corresponding to the metric.
- About the authors
-
- Table of contents (4 chapters)
-
-
Introduction and a Showcase
Pages 1-15
-
Partitions, Weight Functions and Their Hyperbolicity
Pages 17-53
-
Relations of Weight Functions
Pages 55-95
-
Characterization of Ahlfors Regular Conformal Dimension
Pages 97-152
-
Table of contents (4 chapters)
Recommended for you
Bibliographic Information
- Bibliographic Information
-
- Book Title
- Geometry and Analysis of Metric Spaces via Weighted Partitions
- Authors
-
- Jun Kigami
- Series Title
- Lecture Notes in Mathematics
- Series Volume
- 2265
- Copyright
- 2020
- Publisher
- Springer International Publishing
- Copyright Holder
- The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
- eBook ISBN
- 978-3-030-54154-5
- DOI
- 10.1007/978-3-030-54154-5
- Softcover ISBN
- 978-3-030-54153-8
- Series ISSN
- 0075-8434
- Edition Number
- 1
- Number of Pages
- VIII, 164
- Number of Illustrations
- 10 b/w illustrations
- Topics
