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:

": 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)"

It was first proved by for the bounded plane domain and then completely in a more generalized version by. 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 and. The multi (high) - dimensional Suita conjecture fails in non-pseudoconvex domains. This conjecture was proved through the optimal estimation of the Ohsawa–Takegoshi L2 extension theorem.