Interplay Between Loewner and Dirichlet Energies via Conformal Welding and Flow-Lines
Open access
Date
2020-02Type
- Journal Article
ETH Bibliography
yes
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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. Show more
Permanent link
https://doi.org/10.3929/ethz-b-000465668Publication status
publishedExternal links
Journal / series
Geometric and Functional AnalysisVolume
Pages / Article No.
Publisher
SpringerOrganisational unit
09453 - Werner, Wendelin (ehemalig) / Werner, Wendelin (former)
Funding
175505 - Loops, paths and fields (SNF)
Related publications and datasets
Is new version of: http://hdl.handle.net/20.500.11850/395657
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ETH Bibliography
yes
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