On the Loewner energy of a welding homeomorphism

Abstract

To any Jordan curve one may associate a circle homeomorphism $\phi :{\mathbb{S}}^1\to {\mathbb{S}}^1$ via conformal welding. Through this correspondence, the Loewner energy $I^L$, also known as the universal Liouville action, is a Kähler potential for the unique homogeneous Kähler metric on the universal Teichmüller space. Despite this, explicit expressions for $I^L$ in terms of $\phi$ alone do not seem to be available in the literature. In this paper, we obtain such formulas. For this, we introduce an operator ${\mathbf{\Lambda }}_{\phi }$ defined using the Fourier coefficients of the function \[ (z,w) \mapsto \log \left|\frac{φ(z)-φ(w)}{z-w}\right|, \qquad (z,w) \in \mathbb{S}^1 \times \mathbb{S}^1. \] We relate ${\mathbf{\Lambda }}_{\phi }$ to the single-layer potential and composition operator, and prove an analog of the classical Grunsky inequalities for quasisymmetric $\phi$. We show moreover that $\phi$ is Weil--Petersson if and only if ${\mathbf{\Lambda }}_{\phi }$ 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 $\log {\phi }^{\prime }$ and in terms of the composition operator induced by $\phi$.

36 pages