Suita conjecture

In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the theory of the L2 extension. The conjecture states the following: Suita (1972): Let R be an Riemann surface, which admits a nontrivial Green function $G_{R}$. Let $\omega$ be a local coordinate on a neighborhood $V_{z_{0}}$ of $z_{0}\in R$ satisfying $w(z_{0})=0$. Let $\kappa R$ be the Bergman kernel for holomorphic (1, 0) forms on R. We define $B_{R}(z)|dw|^{2}:=\kappa _{R}(z)|_{V_{z_{0}}}$, and $B_{R}(z,{\overline {t}})d\omega \otimes d{\overline {t}}:=\kappa _{R}(z,{\overline {t}})$. Let $c_{\beta }(z)$ be the logarithmic capacity which is locally defined by $c_{\beta }(z_{0}):=\exp \lim _{\xi \to z}(G_{R}(z,z_{0})-\log |\omega (z)|)$ on R. Then, the inequality $(c_{\beta }(z_{0}))^{2}\leq \pi B_{R}(z_{0})$ holds on the every open Riemann surface R, and also, with equality, then $B_{R}\equiv 0$ or, R is conformally equivalent to the unit disc less a (possible) closed set of inner capacity zero.