Fredrik Viklund: The Loewner-Kufarev energy and foliations by Weil-Petersson curves

Abstract

The Loewner-Kufarev equation takes as input a family of measures on the unit circle and outputs a monotone family of simply connected domains by describing the dynamics of the corresponding family of Riemann maps. Vice versa, any continuous and monotone family of domains can be encoded using a family of such measures. This framework is useful to describe and analyze several interesting planar growth processes, including (deterministic) Laplacian growth type models and (random) Schramm-Loewner evolutions (SLEs).

After presenting some background material, I will discuss Loewner-Kufarev evolution corresponding to a particular family of measures satisfying a natural energy condition that arises in the context of large deviations of SLE processes when the kappa parameter is large. The evolving interfaces generated by such a measure form a ``foliating’’ family of non-smooth Jordan curves that exhibits several surprising symmetries. The curves turn out to be Weil-Petersson quasicircles, a very interesting class of curves that has been studied for some time in Teichmuller theory but only recently attracted interest in probability, where they appear for instance in the context of random conformal geometry and coulomb gases in the plane. Based on joint works with Yilin Wang (IHES).