Alice Brolin: Möbius and Loewner energy on curves with corners

Abstract.

The Möbius energy and the Loewner energy are two Möbius invariant quantities defined for Jordan curves. We start by introducing some of the basic properties of these two energies. Both are finite if and only if the curves belong to a class called the Weil–Petersson quasicircles. The Weil–Petersson class does not contain curves with corners. In part motivated by recent work of Johansson and Viklund we introduce regularized versions of both the Möbius and Loewner energy which allow for certain curves with isolated corners. We also look at the derivative of the Loewner energy.