# Ellen Krusell: Reduced Loewner Energies

MSc Thesis Presentation

**Time: **
Wed 2021-06-16 15.00 - 16.00

**Location: **
Zoom, Meeting ID: 637 7637 7139

**Respondent: **
Ellen Krusell

**Supervisor: **
Fredrik Viklund

### Abstract

In 1999 Schramm introduced the one-parameter family of random planar chords known as Schramm-Loewner evolution (SLE(kappa)). More recently, Wang defined a functional on (deterministic) planar chords and loops called Loewner energy. The Loewner energy is the rate function of a large deviation principle on SLE(kappa) as kappa tends to 0. Curves of finite energy are more regular than SLE(kappa) and have several interesting properties. For example, there is a link to Teichmüller theory; the family of finite energy loops coincides with the class of Weil-Petersson quasicircles. In this thesis we study natural generalizations of the chordal Loewner energy. We define a two-sided radial Loewner energy, corresponding to the process of a chordal SLE conditioned to hit a marked interior point. We characterize curves of finite two-sided radial energy and show that there is a unique curve of minimal energy. We then move on to discuss a generalization of the multichordal Loewner energy, introduced by Peltola and Wang, to chords with fused endpoints. First, we construct a multichordal Loewner energy on curves which have not yet reached their respective endpoints, agreeing with the energy defined by Peltola and Wang in the limit. We then generalize this energy to two curves which aim at the same point and define the fused multichordal Loewner energy by taking the limit.