Seminar| Institute of Mathematical Sciences
Time：Monday, June 21th, 2021, 10:00-11:00
Location：R408, IMS
Speaker: Xin Dong, University of Connecticut
Abstract:Without using the $L^2$ extension theorem, we provide a proof of the equality part in Suita's conjecture, which states that for any open Riemann surface $X$ admitting a Green function, the Bergman kernel $K$ and the logarithmic capacity $c_\beta$ coincide at some point if and only if $X$ is biholomorphic to a disc possibly less a closed polar subset. In comparison with Guan and Zhou's proof, our proof only depends on Maitani and Yamaguchi's variation formula and its harmonicity property. After extending the inequality chain $\pi K \geq c_\beta \geq c_B$ by showing $c_B \geq \pi v^{-1}(X)$ on domains in $\mathbb C$, where $c_B$ and $v(\cdot)$ denote the analytic capacity and Euclidean volume, respectively, we explore various rigidity phenomena at the extremal cases when the equalities hold. This talk is based on arXiv: 1807.05537, and a joint work with Treuer and Zhang.