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.

Notes

References

• Błocki, Zbigniew (2013). "Suita conjecture and the Ohsawa-Takegoshi extension theorem". Inventiones Mathematicae. 193 (1): 149–158. Bibcode:2013InMat.193..149B. doi:10.1007/s00222-012-0423-2. S2CID 9209213.
• 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. Vol. 2116. pp. 53–63. doi:10.1007/978-3-319-09477-9_4. ISBN 978-3-319-09476-2.
• Błocki, Zbigniew (2014b). "Cauchy–Riemann meet Monge–Ampère". Bulletin of Mathematical Sciences. 4 (3): 433–480. doi:10.1007/s13373-014-0058-2. S2CID 53582451.
• Błocki, Zbigniew (2017). "Suita Conjecture from the One-dimensional Viewpoint" (PDF). Analysis Meets Geometry. Trends in Mathematics. pp. 127–133. doi:10.1007/978-3-319-52471-9_9. ISBN 978-3-319-52469-6. S2CID 125704662.
• 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. arXiv:1811.02977. doi:10.1007/s12220-019-00343-8. S2CID 119622596.
• Guan, Qi'an; Zhou, Xiangyu (2015). "A solution of an ${\displaystyle L^{2}}$ extension problem with optimal estimate and applications". Annals of Mathematics. 181 (3): 1139–1208. arXiv:1310.7169. doi:10.4007/annals.2015.181.3.6. JSTOR 24523356. S2CID 56205818.
• Nikolov, Nikolai (2015). "Two remarks on the Suita conjecture". Annales Polonici Mathematici. 113: 61–63. arXiv:1411.6601. doi:10.4064/ap113-1-3. S2CID 119147234.
• 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. arXiv:2005.04479. doi:10.1016/j.jmaa.2021.125018. S2CID 218581510.
• Bousfield Classes and Ohkawa's Theorem. Springer Proceedings in Mathematics & Statistics. Vol. 309. 2020. doi:10.1007/978-981-15-1588-0. ISBN 978-981-15-1587-3. S2CID 242194764.
• Ohsawa, Takeo (2017). "On the extension of ${\displaystyle L^{2}}$ 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. Bibcode:1972ArRMA..46..212S. doi:10.1007/BF00252460. S2CID 123118650.

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