Loewner-Kufarev energy and foliations by Weil-Petersson quasicircles

Abstract: We use Loewner-Kufarev equation to describe evolutions of univalent functions and introduce an energy on the driving measure, called Loewner-Kufarev energy. We show that when this energy is finite, the boundaries of the evolving image domains are Weil-Petersson quasicircles which form a foliation of the Riemann sphere. Weil-Petersson quasicircles are studied in Teichmuller theory, geometric function theory, and string theory by both mathematicians and physicists. More than 20 equivalent definitions of this class of Jordan curves are discovered so far. In particular, it is characterized as the class of curves having finite Loewner energy which was also introduced recently. Furthermore, we show that the Loewner-Kufarev energy is dual to the Loewner energy and exhibits remarkable symmetries. Both energies and their duality result are inspired by ideas from the probabilistic theory of Schramm-Loewner evolutions. This is a joint work with Fredrik Viklund (KTH).

References:

The Loewner-Kufarev energy and foliations by Weil-Petersson quasicircles Fredrik Viklund, Yilin Wang (2020) arxiv.org/abs/2012.05771

complex variablesdynamical systems

Audience: researchers in the topic

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