Loewner Energy and Conformal Welding in Complex Analysis

这个研究项目是关于自相似结构的几何。自相似集是在不同尺度下看起来完全相同的集，例如Koch雪花分形曲线。更灵活的保形自相似概念要求一个集合在不同的尺度上看起来大致相同，允许在每个尺度上有界扭曲的变化，例如在Julia集合中看到的。在自然界中观察到的大多数集合都是由涉及随机性的过程产生的，并且满足更弱的“随机自相似性”概念：集合背后的随机机制在所有尺度上都是相同的。该项目旨在理解随机自相似集的基本性质，特别是与我们对共形自相似集的认识有关。树突在这个项目中扮演着一个核心角色，以及如何使用保持角度的坐标变化来描述这种分支随机分形的问题。Loewner微分方程是复杂分析中的一个基本工具，它提供了二维（平面）非自交曲线（如Jordan曲线）和实值驱动函数之间的双射。最近引入的Jordan曲线的“Loewner能量”可以定义为驱动函数的Dirichlet能量。它先验地依赖于曲线的初始点。在这个项目中，将研究与有限能量曲线的正则性有关的几个问题，旨在对能量进行更几何的定义，并证明能量与初始点无关。由于已知有限能量曲线是准共形曲线，因此将采用准共形分析的方法来解决这些问题，特别是全纯变形和保形焊接。对保形焊接的概括导致了通过层合来描述枝晶。这种泛化将在确定性和随机设置中进行探讨。

This research project is concerned with the geometry of self-similar structures. Self-similar sets are sets that look exactly the same at different scales, such as the Koch snowflake fractal curve. The more flexible notion of conformal self-similarity requires a set only to look roughly the same at different scales, allowing for changes of bounded distortion at each scale, such as is seen in Julia sets. Most sets observed in nature arise from processes involving randomness and satisfy the even weaker notion of "stochastic self-similarity": the random mechanism behind the set is the same at all scales. This project aims at an understanding of basic properties of stochastically self-similar sets, particularly in relation to our knowledge of conformally self-similar sets. A central role in the project is played by dendrites and the question of how such branching random fractals can be described using angle-preserving coordinate changes.The Loewner differential equation is a basic tool in complex analysis that provides a bijection between two-dimensional (planar) non-self-crossing curves (such as Jordan curves) and real-valued driving functions. The recently introduced "Loewner energy" of a Jordan curve can be defined as the Dirichlet energy of the driving function. It depends a priori on the initial point of the curve. During this project several questions related to the regularity properties of curves of finite energy will be investigated, aiming at a more geometric definition of the energy and a proof of the independence of energy from the initial point. Since curves of finite energy are known to be quasiconformal curves, the questions will be approached by methods from quasiconformal analysis, particularly holomorphic deformations and conformal welding. A generalization of conformal welding leads to a description of dendrites via laminations. This generalization will be explored both in the deterministic and in the stochastic setting.