Equivalent descriptions of the loewner energy

• 成果类型: Article
• 署名作者: Wang, Yilin
• 署名单位: Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00887-0
• 发表日期: 2019
• 页码: 573-621
• 关键词: weil-petersson; string theory; determinants; geometry

摘要

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 Teichmuller 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 Kahler potential for the Weil-Petersson metric on the Weil-Petersson Teichmuller space.