26 juin 2019

Yilin Wang (ETH Zürich)
The Loewner energy of simple planar curves

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Lieu : Salle 3L8

Résumé : We introduce and study the Loewner energy for simple planar curves and relate this quantity to ideas and concepts coming from random conformal geometry, geometric function theory and Teichmüller theory.
One motivation for the definition of the Loewner energy for chords connecting two boundary points of a simply connected domain is that it arises from the large deviations of Schramm-Loewner evolution (SLE). This provides a probabilistic interpretation of the Loewner energy that allows us to prove its reversibility for any deterministic chord. We generalize the chordal Loewner energy to simple loops on the Riemann sphere and show that the loop energy has a remarkable number of symmetries. We further derive an equivalent characterization of the loop energy using zeta-regularized determinants of Laplacians. This then identifies it with a Kähler potential, introduced by Takhtajan and Teo, of the Weil-Petersson metric on the universal Teichmüller space. In relation to determinants of Laplacians, we derive another measure-theoretic description of the Loewner energy using the Brownian loop measure.

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