On the Loewner energy of a welding homeomorphism

TL;DR

This paper derives explicit formulas for the Loewner energy of a welding homeomorphism using Fourier analysis, operator theory, and Fredholm determinants, connecting geometric and functional analytic perspectives.

Contribution

It introduces an operator based on Fourier coefficients to express the Loewner energy explicitly in terms of the welding homeomorphism, advancing understanding of its analytic structure.

Findings

Expressed Loewner energy as Fredholm determinants and regularized determinants.

Established a criterion linking Weil--Petersson class to Hilbert--Schmidt property of an operator.

Derived formulas involving Dirichlet integrals and composition operators.

Abstract

To any Jordan curve one may associate a circle homeomorphism $\varphi : \mathbb S^1 \to \mathbb S^1$ via conformal welding. Through this correspondence, the Loewner energy $I^L$, also known as the universal Liouville action, is a K\"ahler potential for the unique homogeneous K\"ahler metric on the universal Teichm\"uller space. Despite this, explicit expressions for $I^L$ in terms of $\varphi$ alone do not seem to be available in the literature. In this paper, we obtain such formulas. For this, we introduce an operator ${\bf \Lambda}_\varphi$ defined using the Fourier coefficients of the function \[ (z,w) \mapsto \log \left|\frac{\varphi(z)-\varphi(w)}{z-w}\right|, \qquad (z,w) \in \mathbb{S}^1 \times \mathbb{S}^1. \] We relate ${\bf \Lambda}_\varphi$ to the single-layer potential and composition operator, and prove an analog of the classical Grunsky inequalities for quasisymmetric $\varphi$. We show moreover that $\varphi$ is Weil--Petersson if and only if ${\bf \Lambda}_\varphi$ is Hilbert--Schmidt, and we express $I^L$ as several related Fredholm determinants as well as a regularized Fredholm determinant. We also treat Schatten classes, and we obtain formulas in terms of Dirichlet integrals involving $\log \varphi'$ and in terms of the composition operator induced by $\varphi$.