Equivalent Descriptions of the Loewner Energy

TL;DR

This paper establishes a deep connection between Loewner energy, conformal maps, and Teichmüller theory, revealing that Loewner energy equals the Dirichlet energy of the log-derivative of conformal maps and relates to Weil-Petersson geometry.

Contribution

It proves the equivalence of Loewner energy with the Dirichlet energy of conformal maps and links it to determinants, Teichmüller theory, and the Weil-Petersson class of quasicircles.

Findings

Loewner energy equals the Dirichlet energy of the log-derivative of conformal maps.

Finite Loewner energy loops coincide with Weil-Petersson quasicircles.

Loewner energy is proportional to the universal Liouville action.

Abstract

Loewner's equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map. This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichm\"uller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil-Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a K\"ahler potential for the Weil-Petersson metric on the Weil-Petersson Teichm\"uller space.