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Interplay between the Loewner and Dirichlet energies: Conformal welding and Flow-lines

Date: Tuesday, June 11, 2019 17:30 - 18:30
Speaker: Yilin Wang (ETH Zürich)
Location: Big Seminar room Ground floor / Office Bldg West (I21.EG.101)
Series: Mathematics and CS Seminar
Host: M. Beiglböck, N. Berestycki, L. Erdös, J. Maas
Lab building west seminar room

The Loewner energy defined for Jordan curves is the action functional of SLE (and also its large deviation rate function when the parameter goes to 0). It was shown in a previous work that a Jordan curve has finite energy if and only if it is a Weil-Petersson quasicircle.
In this talk we present identities that relate the Loewner energy to the Dirichlet energy of ambient fields. They are deterministic analogs of both the welding and flow-line couplings of SLEs with the Gaussian free field on the level of action functionals. We deduce also an identity on complex valued fields that combines both welding and flow-line identities. We apply these results to show that the operation of arclength isometric welding of two finite energy domains is sub-additive in the energy and that the energy of equipotentials in a simply connected domain is monotone.
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