Equivalent Descriptions of the Loewner Energy

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 Teichmüller 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 Kähler potential for the Weil-Petersson metric on the Weil-Petersson Teichmüller space.

43 pages, 7 figures. Revised version according to the referee's report, the title is changed

References in corpus (3)

• Exact Solutions for Loewner Evolutions
• The Loewner energy of loops and regularity of driving functions
• Remarks on Loewner Chains Driven by Finite Variation Functions

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