The Loewner chain provides a method for encoding a simple planar curve by a family of uniformizing maps satisfying a differential equation driven by a real-valued function. For instance, choosing Brownian motion for the driving function gives Schramm–Loewner Evolution (SLE). Driving functions with finite Dirichlet energy encode the class of Weil–Petersson quasicircles, which was identified as the semiclassical limit of SLE in a series of works by Yilin Wang. In this talk, we consider the Loewner chain of a simple planar curve under infinitesimal quasiconformal deformations. We provide a variational formula of the Loewner driving function when the Beltrami coefficients are supported away from the curve. As an application, we obtain the first variation of the Loewner energy of a Jordan curve, defined as the Dirichlet energy of the driving function of the curve. This gives another explanation of the identity between the Loewner energy and the universal Liouville action introduced by Takhtajan and Teo. This is joint work with Yilin Wang.

Location: https://ucla.zoom.us/j/181155584   

(Meeting ID: 181 155 584)