The APDE seminar on Monday, 1/31, will be given by Yilin Wang (MIT & MSRI) both in-person (891 Evans) and online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (firstname.lastname@example.org).
Title: Loewner-Kufarev energy and foliation by Weil-Petersson quasicircles
Abstract: Loewner-Kufarev chain encodes a monotone family of univalent functions (injective holomorphic functions on the unit disk) into a driving measure supported on the circle. It is a powerful tool to study the evolution of domains. Example of application includes Schramm-Loewner evolution (SLE), Hasting-Levitov growth model, semigroups of univalent functions, etc.
We introduce the Loewner-Kufarev energy on the driving measure. We show that when the energy is finite, the boundary of the evolving codomains forms a foliation of Weil-Petersson quasicircles (a class of non-smooth Jordan curves with more than 20 equivalent definitions). Moreover, this energy is dual to the leaves’ Loewner energy, which is a fundamental quantity for the Kahler geometry on the space of Jordan curves. This result gives the first example of a complete characterization of rough curves in terms of the Loewner-Kufarev driving measure. Our results, although purely deterministic, are inspired by results from the probabilistic SLE duality theorem. This is a joint work with Fredrik Viklund (KTH).