Shearing process and an example of a bounded support function in $S^0(\mathbb B^2)$

TL;DR

This paper introduces a shearing process for normal Loewner chains and demonstrates the existence of a starlike bounded support function in the two-dimensional ball, contrasting with the one-dimensional case.

Contribution

It develops a shearing process for Loewner chains and constructs a bounded support function in $S^0(\mathbb B^2)$, highlighting new properties in higher dimensions.

Findings

Shearing process produces chains of shears automorphisms.

Existence of a starlike bounded support function in $S^0(\mathbb B^2)$.

Contrasts with the one-dimensional case.

Abstract

We introduce a process, that we call "shearing", which for any given normal Loewner chain produces a normal Loewner chain made of shears automorphisms. As an application, and in stringent contrast to the one-dimensional case, we prove the existence of a starlike bounded function in the class $S^0$ of the ball $\mathbb B^2$ (in fact the restriction of a shear automorphism of $\mathbb C^2$) which is a support point for a linear continuous functional.