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Suita conjecture

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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 L extension. The conjecture states the following:

Suita (1972): Let R be an Riemann surface, which admits a nontrivial Green function G R {\displaystyle G_{R}} . Let ω {\displaystyle \omega } be a local coordinate on a neighborhood V z 0 {\displaystyle V_{z_{0}}} of z 0 ∈ R {\displaystyle z_{0}\in R} satisfying w ( z 0 ) = 0 {\displaystyle w(z_{0})=0} . Let κ R {\displaystyle \kappa R} be the Bergman kernel for holomorphic (1, 0) forms on R. We define B R ( z ) | d w | 2 := κ R ( z ) | V z 0 {\displaystyle B_{R}(z)|dw|^{2}:=\kappa _{R}(z)|_{V_{z_{0}}}} , and B R ( z , t ¯ ) d ω ⊗ d t ¯ := κ R ( z , t ¯ ) {\displaystyle B_{R}(z,{\overline {t}})d\omega \otimes d{\overline {t}}:=\kappa _{R}(z,{\overline {t}})} . Let c β ( z ) {\displaystyle c_{\beta }(z)} be the logarithmic capacity which is locally defined by c β ( z 0 ) := exp ⁡ lim ξ → z ( G R ( z , z 0 ) − log ⁡ | ω ( z ) | ) {\displaystyle c_{\beta }(z_{0}):=\exp \lim _{\xi \to z}(G_{R}(z,z_{0})-\log |\omega (z)|)} on R. Then, the inequality ( c β ( z 0 ) ) 2 ≤ π B R ( z 0 ) {\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 B R ≡ 0 {\displaystyle B_{R}\equiv 0} or, R is conformally equivalent to the unit disc less a (possible) closed set of inner capacity zero.

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. This conjecture was proved through the optimal estimation of the Ohsawa–Takegoshi L extension theorem.

Notes

  1. Guan & Zhou (2015)
  2. Nikolov (2015), Nikolov & Thomas (2021)

References


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