On the geometry of locally growing Loewner chains

Abstract

Loewner chains are ubiquitous in the theory of slit mappings, and hence in the study of bounded conformal maps. They have attracted new interest in the past decades through their applications to statistical physics and fractal geometry, particularly in contexts involving randomness. In this article, we delve into topological features of the growing hulls obtained from Loewner chains with a general local growth property, inspired by the classical works of Loewner and Pommerenke. We first revisit Loewner's theorem, associating to each locally growing collection of hulls a real-valued driving function W, possibly discontinuous. We then investigate the points chronologically added to the growing hulls, which may be part of a simply connected swallowed ``bubble'', or a compact connected boundary set. For continuous driving functions, the Loewner chain can often be associated with a continuous curve (dubbed ``generating curve''). Motivated by this, we introduce a more general notion of a ``generating function'' for the Loewner chain, and characterize when there exists such a function {\eta} (which can be continuous, c\`adl\`ag, c\`agl\`ad, or neither). We then investigate the necessity of left and right limits for {\eta} from the point of view of the topology of the growing hulls. We find in particular that left-continuity implies path-connectedness and local connectedness of the hulls, as well as the existence of right limits, whereas failure of left-continuity leads to pathological boundary behavior.