Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines
Name
39_2020_Article_521.pdf
Size
602.88 KB
Format
Adobe PDF
Checksum (MD5)
3f95d7e3740af1d997ccdf88c0c849ab
Author(s)
Viklund, Fredrik
Wang, Yilin
Date Issued
February 13, 2020
Publisher
Springer International Publishing
Version
Final published version
Abstract
Abstract
The Loewner energy of a Jordan curve is the Dirichlet energy of its Loewner driving term. It is finite if and only if the curve is a Weil–Petersson quasicircle. In this paper, we describe cutting and welding operations on finite Dirichlet energy functions defined in the plane, allowing expression of the Loewner energy in terms of Dirichlet energy dissipation. We show that the Loewner energy of a unit vector field flow-line is equal to the Dirichlet energy of the harmonically extended winding. We also give an identity involving a complex-valued function of finite Dirichlet energy that expresses the welding and flow-line identities simultaneously. As applications, we prove that arclength isometric welding of two domains is sub-additive in the energy, and that the energy of equipotentials in a simply connected domain is monotone. Our main identities can be viewed as action functional analogs of both the welding and flow-line couplings of Schramm–Loewner evolution curves with the Gaussian free field.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00039-020-00521-9
