Yan Luo: Analytic and geometric approach towards a generalized Loewner potential

Abstract: We two functionals on the space of annuli as cadidates of a generalized Loewner potential. The first one is obtained from the zeta-regularized determinants of Laplacians. Using the Polyakov-Alvarez formula, we get an explicit formula of it as Dirichlet energy of pre-Schwarzians. We also have an equation expressing the generalized Loewner potential as a sum of the Loewner energy of the two boundary curves with a renormalized Brownian loop term. Another approach is the renormalized volume of Epstein-Poincaré surfaces over the annulus. The two functionals can be related via the quasiconformal variations preserving the modulus. Part of this talk is based on joint work with Sid Maibach.