The $p$-Weil-Petersson Teichm\"uller space and the quasiconformal extension of curves

TL;DR

This paper establishes a connection between $p$-Weil-Petersson curves and $p$-Besov spaces, showing that a heat kernel-based Beurling-Ahlfors extension provides a holomorphic map and a real-analytic section for the Teichmüller projection.

Contribution

It introduces a new holomorphic extension method for $p$-Weil-Petersson curves using heat kernel techniques, providing a global real-analytic section for the Teichmüller projection.

Findings

The Beurling-Ahlfors extension via heat kernel is holomorphic on a domain of $p$-Besov space.

A global real-analytic section for the Teichmüller projection is constructed.

The correspondence between $p$-Weil-Petersson curves and $p$-Besov spaces is established.

Abstract

We consider the correspondence between the space of $p$-Weil-Petersson curves $\gamma$ on the plane and the $p$-Besov space of $u=\log \gamma'$ on the real line for $p >1$. We prove that the variant of the Beurling-Ahlfors extension defined by using the heat kernel yields a holomorphic map for $u$ on a domain of the $p$-Besov space to the space of $p$-integrable Beltrami coefficients. This in particular gives a global real-analytic section for the Teichm\"uller projection from the space of $p$-integrable Beltrami coefficients to the $p$-Weil-Petersson Teichm\"uller space.