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 ${\displaystyle G_{R}}$. Let ${\displaystyle \omega }$ be a local coordinate on a neighborhood ${\displaystyle V_{z_{0}}}$ of ${\displaystyle z_{0}\in R}$ satisfying ${\displaystyle w(z_{0})=0}$. Let ${\displaystyle \kappa R}$ be the Bergman kernel for holomorphic (1, 0) forms on R. We define ${\displaystyle B_{R}(z)|dw|^{2}:=\kappa _{R}(z)|_{V_{z_{0}}}}$, and ${\displaystyle B_{R}(z,{\overline {t}})d\omega \otimes d{\overline {t}}:=\kappa _{R}(z,{\overline {t}})}$ . Let ${\displaystyle c_{\beta }(z)}$ be the logarithmic capacity which is locally defined by ${\displaystyle c_{\beta }(z_{0}):=\exp \lim _{\xi \to z}(G_{R}(z,z_{0})-\log |\omega (z)|)}$ on R. Then, the inequality ${\displaystyle (c_{\beta }(z_{0}))^{2}\leq \pi B_{R}(z_{0})}$ holds on the every open Riemann surface R, and also, with equality, then ${\displaystyle B_{R}\equiv 0}$ or, R is conformally equivalent to the unit disc less a (possible) closed set of inner capacity zero.[1]

It was first proved by Błocki (2013) for the bounded plane domain and then completely in a more generalized version by Guan & Zhou (2015). Also, another proof of the Suita conjecture and some examples of its generalization to several complex variables (the multi (high) - dimensional Suita conjecture) were given in Błocki (2014a) and Błocki & Zwonek (2020). The multi (high) - dimensional Suita conjecture fails in non-pseudoconvex domains.[2] This conjecture was proved through the optimal estimation of the Ohsawa–Takegoshi L² extension theorem.