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 [math]\displaystyle{ G_R }[/math]. Let [math]\displaystyle{ \omega }[/math] be a local coordinate on a neighborhood [math]\displaystyle{ V_{z_0} }[/math] of [math]\displaystyle{ z_0 \in R }[/math] satisfying [math]\displaystyle{ w(z_0) = 0 }[/math]. Let [math]\displaystyle{ \kappa R }[/math] be the Bergman kernel for holomorphic (1, 0) forms on R. We define [math]\displaystyle{ B_{R}(z)|dw|^2 := \kappa_{R}(z)|_{V_{z_0}} }[/math], and [math]\displaystyle{ B_{R}(z, \overline{t})d\omega \otimes d\overline{t} := \kappa_{R}(z,\overline{t}) }[/math] . Let [math]\displaystyle{ c_{\beta}(z) }[/math] be the logarithmic capacity which is locally defined by [math]\displaystyle{ c_{\beta}(z_0) := \exp \lim_{\xi \to z} (G_{R}(z, z_{0}) -\log |\omega(z)|) }[/math] on R. Then, the inequality [math]\displaystyle{ (c_{\beta}(z_{0}))^2 \leq \pi B_{R}(z_0) }[/math] holds on the every open Riemann surface R, and also, with equality, then [math]\displaystyle{ B_{R} \equiv 0 }[/math] 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). 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). 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.

References

• Błocki, Zbigniew (2013). "Suita conjecture and the Ohsawa-Takegoshi extension theorem". Inventiones Mathematicae 193 (1): 149–158. doi:10.1007/s00222-012-0423-2. Bibcode: 2013InMat.193..149B. 
• Błocki, Zbigniew (2014a). "A Lower Bound for the Bergman Kernel and the Bourgain-Milman Inequality". Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2011-2013. Lecture Notes in Mathematics. 2116. pp. 53–63. doi:10.1007/978-3-319-09477-9_4. ISBN 978-3-319-09476-2. https://books.google.com/books?id=t6_CBAAAQBAJ&pg=PA52. 
• Błocki, Zbigniew (2014b). "Cauchy–Riemann meet Monge–Ampère". Bulletin of Mathematical Sciences 4 (3): 433–480. doi:10.1007/s13373-014-0058-2. 
• Błocki, Zbigniew (2017). "Suita Conjecture from the One-dimensional Viewpoint". Analysis Meets Geometry. Trends in Mathematics. pp. 127–133. doi:10.1007/978-3-319-52471-9_9. ISBN 978-3-319-52469-6. http://gamma.im.uj.edu.pl/~blocki/publ/mikael.pdf. 
• Błocki, Zbigniew; Zwonek, Włodzimierz (2020). "Generalizations of the Higher Dimensional Suita Conjecture and Its Relation with a Problem of Wiegerinck". The Journal of Geometric Analysis 30 (2): 1259–1270. doi:10.1007/s12220-019-00343-8. 
• Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an [math]\displaystyle{ L^2 }[/math] extension problem with optimal estimate and applications". Annals of Mathematics 181 (3): 1139–1208. doi:10.4007/annals.2015.181.3.6. 
• Nikolov, Nikolai (2015). "Two remarks on the Suita conjecture". Annales Polonici Mathematici 113: 61–63. doi:10.4064/ap113-1-3. 
• Nikolov, Nikolai; Thomas, Pascal J. (2021). "Growth of Sibony metric and Bergman kernel for domains with low regularity". Journal of Mathematical Analysis and Applications 499: 125018. doi:10.1016/j.jmaa.2021.125018. 
• Bousfield Classes and Ohkawa's Theorem. Springer Proceedings in Mathematics & Statistics. 309. 2020. doi:10.1007/978-981-15-1588-0. ISBN 978-981-15-1587-3. https://books.google.com/books?id=ssPXDwAAQBAJ&pg=PA426. 
• Ohsawa, Takeo (2017). "On the extension of [math]\displaystyle{ L^2 }[/math] holomorphic functions VIII — a remark on a theorem of Guan and Zhou". International Journal of Mathematics 28 (9). doi:10.1142/S0129167X17400055. 
• Suita, Nobuyuki (1972). "Capacities and kernels on Riemann surfaces". Archive for Rational Mechanics and Analysis 46 (3): 212–217. doi:10.1007/BF00252460. Bibcode: 1972ArRMA..46..212S.