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Lecture Notes in Mathematics

Geometry and Analysis of Metric Spaces via Weighted Partitions

Authors: Kigami, Jun

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  • 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)
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  • ISBN 978-3-030-54154-5
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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:

  1. It is shown that a weight corresponds to a metric if and only if the associated weighted graph is Gromov hyperbolic.
  2. 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.
  3. 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.

 These notes should interest researchers and PhD students working in conformal geometry, analysis on metric spaces, and related areas.


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Table of contents (4 chapters)

Table of contents (4 chapters)

Buy this book

eBook $39.99
price for USA in USD
  • ISBN 978-3-030-54154-5
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Geometry and Analysis of Metric Spaces via Weighted Partitions
Authors
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