The VMO-Teichm\"uller space and the variant of Beurling-Ahlfors extension by heat kernel

TL;DR

This paper constructs a real-analytic structure on the VMO-Teichmüller space using a heat kernel-based Beurling-Ahlfors extension, establishing its Banach manifold properties and contraction mappings.

Contribution

It introduces a real-analytic section for the Teichmüller projection onto VMO-Teichmüller space using heat kernel extension, and proves the space's Banach manifold structure and contraction properties.

Findings

Established a real-analytic section for the Teichmüller projection.

Proved VMO-Teichmüller space has a real Banach manifold structure.

Demonstrated the existence of a real-analytic contraction mapping.

Abstract

We give a real-analytic section for the Teichm\"uller projection onto the VMO-Teichm\"uller space by using the variant of Beurling-Ahlfors extension by heat kernel introduced by Fefferman, Kenig and Pipher in 1991. Based on this result, we prove that the VMO-Teichm\"uller space can be endowed with a real Banach manifold structure that is real-analytically equivalent to its complex Banach manifold structure. We also obtain that the VMO-Teichm\"uller space admits a real-analytic contraction mapping.