Geometry and Analysis of Metric Spaces via Weighted Partitions

Author: Prof. Jun Kigami

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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.

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

Frontmatter

Chapter 1. Introduction and a Showcase

Abstract

Successive divisions of a space have played important roles in many areas of mathematics. One of the simplest examples is the binary division of the unit interval [0, 1] shown in Fig. 1.1. Let K _ϕ = [0, 1] and divide K _ϕ in half as $K_0 = [0, \frac 12]$ and $K_1 = [\frac 12, 1]$. Next, K ₀ and K ₁ are divided in half again and yield K _ij for each (i, j) ∈{0, 1}². Repeating this procedure, we obtain $\{K_{{i}_1\ldots {i}_{m}}\}_{i_1, \ldots , i_m \in \{0, 1\}}$ satisfying

$$\displaystyle K_{{i}_1\ldots {i}_{m}} = K_{{i}_1\ldots {i}_{m}0} \cup K_{{i}_1\ldots {i}_{m}1}$$

for any m ≥ 0 and i ₁…i _m ∈{0, 1}^m. In this example, there are two notable properties.

Jun Kigami

Chapter 2. Partitions, Weight Functions and Their Hyperbolicity

Abstract

In this section, we review basic notions and notations on a tree with a reference point.

Jun Kigami

Chapter 3. Relations of Weight Functions

Abstract

In this section, we define the notion of bi-Lipschitz equivalence of weight functions. Originally the definition, Definition 3.1.1, only concerns the tree structure $(T, \mathcal {A}, \phi )$ and has nothing to do with a partition of a space.

Jun Kigami

Chapter 4. Characterization of Ahlfors Regular Conformal Dimension

Abstract

In this section, we present a sufficient condition for the existence of an adapted metric to a given weight function. The sufficient condition obtained in this section will be used to construct an Ahlfors regular metric later.

Jun Kigami

Backmatter

Title: Geometry and Analysis of Metric Spaces via Weighted Partitions
Author: Prof. Jun Kigami
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-54154-5
Print ISBN: 978-3-030-54153-8
DOI: https://doi.org/10.1007/978-3-030-54154-5

Springer Professional

Geometry and Analysis of Metric Spaces via Weighted Partitions

About this book

Table of Contents

Frontmatter

Chapter 1. Introduction and a Showcase

Chapter 2. Partitions, Weight Functions and Their Hyperbolicity

Chapter 3. Relations of Weight Functions

Chapter 4. Characterization of Ahlfors Regular Conformal Dimension

Backmatter

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