The Loewner energy of chords in simply connected domain

par Yilin Wang (ETH Zurich)

jeudi 20 oct. 2016, 14:30 → 15:30 Europe/Paris

salle 435 (UMPA)

We study some features of the energy of a deterministic chordal Loewner chain, which is defined as the Dirichlet energy of its driving function in a very directional way and is conformally invariant. Using an interpretation of this energy as a large deviation rate function for SLE_k as k goes to 0, we show that the energy of a deterministic curve from one boundary point A of a simply connected domain D to another boundary point B, is equal to the energy of its time-reversal i.e. of the same curve but viewed as going from B to A in D. In particular it measures how far does the chord differ from the hyperbolic geodesic. The relation between the energy of the curve with its regularity will be discussed, some questions are still open. If time allows, I will present the Loewner energy for loops on the Riemann sphere, and open questions related to it as well.