Abstract
For a once-punctured complex torus, we compare the Bergman kernel and the fundamental metric, by constructing explicitly the Evans-Selberg potential and discussing its asymptotic behaviors. This work aims to generalize the Suita type results to potential-theoretically parabolic Riemann surfaces.
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Notes
- 1.
The author apologizes for several mistakes contained in [3].
References
B. Berndtsson, L. Lempert, A proof of the Ohsawa-Takegoshi theorem with sharp estimates. J. Math. Soc. Jpn. 68 (4), 1461–1472 (2016)
Z. Błocki, Suita conjecture and the Ohsawa-Takegoshi extension theorem. Invent. Math. 193 (1), 149–158 (2013)
R.X. Dong, Suita conjecture for a complex torus. Chin. Ann. Math. 35A (1), 101–108 (2014) (in Chinese); Translation in Chin. J. Contemp. Math. 35 (1), 83–88 (2014)
R.X. Dong, Evans-Selberg potential on planar domains. Proc. Jpn. Acad. 93 (4), 23–26 (2017); Suita conjecture for a punctured torus, in New Trends in Analysis and Interdisciplinary Applications: Selected Contributions of the 10th ISAAC Congress (Macau 2015). Trends in Mathematics (Birkhäuser/Springer, Basel, 2017), pp. 187–193
Q.-A. Guan, X.-Y. Zhou, A solution of an L 2 extension problem with optimal estimate and applications. Ann. Math. 181 (3), 1139–1208 (2015)
J.D. McNeal, D. Varolin, L 2 estimates for the \(\bar{\partial }\) operator. Bull. Math. Sci. 5, 179–249 (2015)
T. Ohsawa, Addendum to “On the Bergman kernel of hyperconvex domains”. Nagoya Math. J. 137, 145–148 (1995)
T. Ohsawa, L 2 Approaches in Several Complex Variables. Springer Monographs in Mathematics (Springer, Tokyo, 2015)
H. Ooguri, International Lectures on Frontier Physics 1 (2010). Online available at http://ocw.u-tokyo.ac.jp/lecture_files/sci_03/9/notes/en/ooguri09.pdf
L. Sario, M. Nakai, Classification Theory of Riemann surfaces. Grundlehren Math. Wiss (Springer, Berlin, 1970)
L. Sario, K. Noshiro, Value Distribution Theory (Van Nostrand, Princeton, 1966)
N. Suita, Capacities and kernels on Riemann surfaces. Arch. Ration. Mech. Anal. 46, 212–217 (1972)
Acknowledgements
The author expresses his gratitude to Prof. T. Ohsawa for his guidance and to Prof. H. Umemura for the communications on elliptic functions. He also thanks H. Fujino and X. Liu for the discussions.
This work is supported by the Ideas Plus grant 0001/ID3/2014/63 of the Polish Ministry of Science and Higher Education, KAKENHI and the Grant-in-Aid for JSPS Fellows (No. 15J05093).
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Dong, R.X. (2017). Suita Conjecture for a Punctured Torus. In: Dang, P., Ku, M., Qian, T., Rodino, L. (eds) New Trends in Analysis and Interdisciplinary Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-48812-7_25
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DOI: https://doi.org/10.1007/978-3-319-48812-7_25
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