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.