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Equivalent descriptions of the loewner energy

  • Yilin WangEmail author
Article
  • 460 Downloads

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.

Mathematics Subject Classification

Primary 30C55 30C62 Secondary 30F60 11M36 

Notes

Acknowledgements

I would like to thank Wendelin Werner for numerous inspiring discussions as well as his help during the preparation of the manuscript. I also thank Steffen Rohde, Yuliang Shen, Lee-Peng Teo, Thomas Kappeler, Alexis Michelat and Tristan Rivière for helpful discussions, and the referee for many constructive comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsETHZürichSwitzerland

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