Geometry and Analysis of Metric Spaces via Weighted Partitions

Jun Kigami

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English

Springer Nature Switzerland AG
17 November 2020

Calculus & mathematical analysis; Integral calculus & equations; Geometry; Non-Euclidean geometry; Topology

Series: Lecture Notes in Mathematics

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.

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

By: Jun Kigami
Imprint: Springer Nature Switzerland AG
Country of Publication: Switzerland
Edition: 1st ed. 2020
Volume: 2265
Dimensions: Height: 235mm, Width: 155mm,
Weight: 454g
ISBN: 9783030541538
ISBN 10: 3030541533
Series: Lecture Notes in Mathematics
Pages: 164
Publication Date: 17 November 2020
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

Reviews for Geometry and Analysis of Metric Spaces via Weighted Partitions

“The monograph is well-written and concerns a novel idea which has great potential to become a major concept in areas such as fractal geometry and dynamical systems theory. It is written at the level of graduate students and for researchers interested in the aforementioned areas.” (Peter Massopust, zbMATH 1455.28001, 2021)