Estimates of the Bergman kernel on Teichmüller space
HTML articles powered by AMS MathViewer
- by Guangming Hu and Hideki Miyachi;
- Proc. Amer. Math. Soc. 151 (2023), 2895-2905
- DOI: https://doi.org/10.1090/proc/15580
- Published electronically: April 6, 2023
- HTML | PDF | Request permission
Abstract:
In this paper, a comparison between the Bergman kernel form and the pushforward measure of the Masur-Veech measure on the Teichmüller space of genus $g\ge 2$ is obtained.References
- L. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975–978. MR 148896, DOI 10.1090/S0002-9939-1962-0148896-1
- J. C. Álvarez Paiva and A. C. Thompson, Volumes on normed and Finsler spaces, A sampler of Riemann-Finsler geometry, Math. Sci. Res. Inst. Publ., vol. 50, Cambridge Univ. Press, Cambridge, 2004, pp. 1–48. MR 2132656, DOI 10.4171/prims/123
- Jayadev Athreya, Alexander Bufetov, Alex Eskin, and Maryam Mirzakhani, Lattice point asymptotics and volume growth on Teichmüller space, Duke Math. J. 161 (2012), no. 6, 1055–1111. MR 2913101, DOI 10.1215/00127094-1548443
- Kazuo Azukawa, The invariant pseudometric related to negative plurisubharmonic functions, Kodai Math. J. 10 (1987), no. 1, 83–92. MR 879385, DOI 10.2996/kmj/1138037363
- Lipman Bers, Simultaneous uniformization, Bull. Amer. Math. Soc. 66 (1960), 94–97. MR 111834, DOI 10.1090/S0002-9904-1960-10413-2
- Lipman Bers, Correction to “Spaces of Riemann surfaces as bounded domains”, Bull. Amer. Math. Soc. 67 (1961), 465–466. MR 130972, DOI 10.1090/S0002-9904-1961-10637-X
- J. Bland and Ian Graham, On the Hausdorff measures associated to the Carathéodory and Kobayashi metrics, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 12 (1985), no. 4, 503–514 (1986). MR 848839
- Zbigniew Blocki, A lower bound for the Bergman kernel and the Bourgain-Milman inequality, Geometric aspects of functional analysis, Lecture Notes in Math., vol. 2116, Springer, Cham, 2014, pp. 53–63. MR 3364678, DOI 10.1007/978-3-319-09477-9_{4}
- Zbigniew Błocki and Włodzimierz Zwonek, Estimates for the Bergman kernel and the multidimensional Suita conjecture, New York J. Math. 21 (2015), 151–161. MR 3318425
- Herbert Busemann, Intrinsic area, Ann. of Math. (2) 48 (1947), 234–267. MR 20626, DOI 10.2307/1969168
- Bo-Yong Chen, The Bergman metric on Teichmüller space, Internat. J. Math. 15 (2004), no. 10, 1085–1091. MR 2106264, DOI 10.1142/S0129167X04002697
- Seán Dineen, The Schwarz lemma, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1989. Oxford Science Publications. MR 1033739
- Spencer Dowdall, Moon Duchin, and Howard Masur, Statistical hyperbolicity in Teichmüller space, Geom. Funct. Anal. 24 (2014), no. 3, 748–795. MR 3213829, DOI 10.1007/s00039-014-0265-8
- Clifford J. Earle and Irwin Kra, On holomorphic mappings between Teichmüller spaces, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York-London, 1974, pp. 107–124. MR 430319
- Frederick P. Gardiner, Teichmüller theory and quadratic differentials, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1987. A Wiley-Interscience Publication. MR 903027
- Y. Imayoshi and M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. Translated and revised from the Japanese by the authors. MR 1215481, DOI 10.1007/978-4-431-68174-8
- M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France 113 (1985), no. 2, 231–240 (English, with French summary). MR 820321, DOI 10.24033/bsmf.2029
- Shoshichi Kobayashi, Geometry of bounded domains, Trans. Amer. Math. Soc. 92 (1959), 267–290. MR 112162, DOI 10.1090/S0002-9947-1959-0112162-5
- Irwin Kra, On Teichmüller spaces for finitely generated Fuchsian groups, Amer. J. Math. 91 (1969), 67–74. MR 257351, DOI 10.2307/2373269
- Samuel L. Krushkal, The Green function of Teichmüller spaces with applications, Bull. Amer. Math. Soc. (N.S.) 27 (1992), no. 1, 143–147. MR 1142683, DOI 10.1090/S0273-0979-1992-00294-X
- László Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), no. 4, 427–474 (French, with English summary). MR 660145, DOI 10.24033/bsmf.1948
- Howard Masur, Interval exchange transformations and measured foliations, Ann. of Math. (2) 115 (1982), no. 1, 169–200. MR 644018, DOI 10.2307/1971341
- Hideki Miyachi, Pluripotential theory on Teichmüller space I: Pluricomplex Green function, Conform. Geom. Dyn. 23 (2019), 221–250. MR 4028456, DOI 10.1090/ecgd/340
- Nikolai Nikolov and Pascal J. Thomas, Comparison of the Bergman kernel and the Carathéodory-Eisenman volume, Proc. Amer. Math. Soc. 147 (2019), no. 11, 4915–4919. MR 4011523, DOI 10.1090/proc/14604
- Takeo Ohsawa, $L^2$ approaches in several complex variables, Springer Monographs in Mathematics, Springer, Tokyo, 2018. Towards the Oka-Cartan theory with precise bounds; Second edition of [ MR3443603]. MR 3887636, DOI 10.1007/978-4-431-56852-0
- H. L. Royden, Automorphisms and isometries of Teichmüller space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969) Ann. of Math. Stud., No. 66, Princeton Univ. Press, Princeton, NJ, 1971, pp. 369–383. MR 288254
- Nobuyuki Suita, Capacities and kernels on Riemann surfaces, Arch. Rational Mech. Anal. 46 (1972), 212–217. MR 367181, DOI 10.1007/BF00252460
- William A. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2) 115 (1982), no. 1, 201–242. MR 644019, DOI 10.2307/1971391
Bibliographic Information
- Guangming Hu
- Affiliation: College of Science, Jinling Institute of Technology, Nanjing 211169, People’s Republic of China
- Email: 18810692738@163.com
- Hideki Miyachi
- Affiliation: School of Mathematics and Physics, College of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, Ishikawa 920-1192, Japan
- MR Author ID: 650573
- ORCID: 0000-0003-4318-9539
- Email: miyachi@se.kanazawa-u.ac.jp
- Received by editor(s): August 15, 2020
- Received by editor(s) in revised form: February 5, 2021
- Published electronically: April 6, 2023
- Additional Notes: The first author was partially supported by PhD research startup foundation of Jinling Institute Technology Grant Number jit-b-202011 and jit-fhxm-2018
The second author was partially supported by JSPS KAKENHI Grant Numbers 20H01800, 16H03933, 17H02843 - Communicated by: Filippo Bracci
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 2895-2905
- MSC (2020): Primary 32G05, 32G15, 32U35, 32A25, 32F45; Secondary 53C60
- DOI: https://doi.org/10.1090/proc/15580
- MathSciNet review: 4579365