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Reversibility of the Loewner energy

Yilin Wang, ETH Zurich
Tuesday, January 17, 2017 - 1:30pm to 3:30pm
Padelford C-401

Loewner introduced in 1923 a way to encode non-self-intersecting two-dimensional paths by
a one-dimensional continuous driving function. This method is an important tool in the
proof of Bieberbach conjecture and gives rise to a natural family of random curves, SLEs.
We define the Loewner energy of a deterministic chord as the Dirichlet energy of its driving
function. It is a priori defined for a directed chord from one boundary point A of a simply
connected domain D to another boundary point B, and is conformally invariant. Using an
interpretation of this energy as a large deviation rate function for \$SLEκ\$ as \$κ\$ goes to 0, we
show that the energy is reversible, i.e. it remains the same if the chord is viewed as going
from B to A in D. In consequence, the Loewner energy measures how far does the chord
differ from the hyperbolic geodesic.

The first part of the talk consists of a brief overview of the Loewner theory and the Loewner
energy. I will try to convey an intuition on what the Loewner energy measures. In the
second part, I will introduce SLE, large deviation principle and how are they related to the
Loewner energy and its reversibility. I will outline the proof. If time allows, I will present
the Loewner energy for loops on the Riemann sphere, and open questions related to it.

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