Unbounded visibility domains, the end compactification, and applications

Bharali, Gautam; Zimmer, Andrew

doi:10.1090/tran/8944

Unbounded visibility domains, the end compactification, and applications
HTML articles powered by AMS MathViewer

by Gautam Bharali and Andrew Zimmer;

Trans. Amer. Math. Soc. 376 (2023), 5949-5988

DOI: https://doi.org/10.1090/tran/8944

Published electronically: May 31, 2023

HTML | PDF | Request permission

Abstract:

In this paper we study when the Kobayashi distance on a Kobayashi hyperbolic domain has certain visibility properties, with a focus on unbounded domains. “Visibility” in this context is reminiscent of visibility, seen in negatively curved Riemannian manifolds, in the sense of Eberlein–O’Neill. However, we do not assume that the domains studied are Cauchy-complete with respect to the Kobayashi distance, as this is hard to establish for domains in $\mathbb {C}^d$, $d \geq 2$. We study the various ways in which this property controls the boundary behavior of holomorphic maps. Among these results is a Carathéodory-type extension theorem for biholomorphisms between planar domains—notably: between infinitely-connected domains. We also explore connections between our visibility property and Gromov hyperbolicity of the Kobayashi distance.

References

Marco Abate, Horospheres and iterates of holomorphic maps, Math. Z. 198 (1988), no. 2, 225–238. MR 939538, DOI 10.1007/BF01163293
Marco Abate, Iteration theory of holomorphic maps on taut manifolds, Research and Lecture Notes in Mathematics. Complex Analysis and Geometry, Mediterranean Press, Rende, 1989. MR 1098711
Marco Abate, Iteration theory, compactly divergent sequences and commuting holomorphic maps, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 2, 167–191. MR 1129300
Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 450957
Marco Abate and Peter Heinzner, Holomorphic actions on contractible domains without fixed points, Math. Z. 211 (1992), no. 4, 547–555. MR 1191095, DOI 10.1007/BF02571445
Marco Abate and Jasmin Raissy, Wolff-Denjoy theorems in nonsmooth convex domains, Ann. Mat. Pura Appl. (4) 193 (2014), no. 5, 1503–1518. MR 3262645, DOI 10.1007/s10231-013-0341-y
Zoltán M. Balogh and Mario Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000), no. 3, 504–533. MR 1793800, DOI 10.1007/s000140050138
Filippo Bracci, Manuel D. Contreras, and Santiago Díaz-Madrigal, Continuous semigroups of holomorphic self-maps of the unit disc, Springer Monographs in Mathematics, Springer, Cham, [2020] ©2020. MR 4252032, DOI 10.1007/978-3-030-36782-4
A. F. Beardon, The dynamics of contractions, Ergodic Theory Dynam. Systems 17 (1997), no. 6, 1257–1266. MR 1488316, DOI 10.1017/S0143385797086434
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Monika Budzyńska, Tadeusz Kuczumow, and Simeon Reich, Theorems of Denjoy-Wolff type, Ann. Mat. Pura Appl. (4) 192 (2013), no. 4, 621–648. MR 3081638, DOI 10.1007/s10231-011-0240-z
Gautam Bharali and Anwoy Maitra, A weak notion of visibility, a family of examples, and Wolff-Denjoy theorems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 22 (2021), no. 1, 195–240. MR 4288653
Filippo Bracci, Nikolai Nikolov, and Pascal J. Thomas, Visibility of Kobayashi geodesics in convex domains and related properties, Math. Z. 301 (2022), no. 2, 2011–2035. MR 4418345, DOI 10.1007/s00209-022-02978-w
Filippo Bracci, Personal communication, 2022.
Gautam Bharali and Andrew Zimmer, Goldilocks domains, a weak notion of visibility, and applications, Adv. Math. 310 (2017), 377–425. MR 3620691, DOI 10.1016/j.aim.2017.02.005
Chin-Huei Chang, M. C. Hu, and Hsuan-Pei Lee, Extremal analytic discs with prescribed boundary data, Trans. Amer. Math. Soc. 310 (1988), no. 1, 355–369. MR 930081, DOI 10.1090/S0002-9947-1988-0930081-7
E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge University Press, Cambridge, 1966. MR 231999, DOI 10.1017/CBO9780511566134
Vikramjeet Singh Chandel, Anwoy Maitra, and Amar Deep Sarkar, Notions of visibility with respect to the Kobayashi distance: comparison and applications, Preprint, arXiv:2111.00549, 2021.
Arnaud Denjoy, Les fonctionnelles linéaires, C. R. Acad. Sci. Paris 250 (1960), 250–255 (French). MR 109869
Tushar Das, David Simmons, and Mariusz Urbański, Geometry and dynamics in Gromov hyperbolic metric spaces, Mathematical Surveys and Monographs, vol. 218, American Mathematical Society, Providence, RI, 2017. With an emphasis on non-proper settings. MR 3558533, DOI 10.1090/surv/218
Matteo Fiacchi, Gromov hyperbolicity of pseudoconvex finite type domains in $\Bbb C^2$, Math. Ann. 382 (2022), no. 1-2, 37–68. MR 4377298, DOI 10.1007/s00208-020-02135-w
Franc Forstnerič and Jean-Pierre Rosay, Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann. 279 (1987), no. 2, 239–252. MR 919504, DOI 10.1007/BF01461721
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
Hervé Gaussier and Harish Seshadri, On the Gromov hyperbolicity of convex domains in $\Bbb C^n$, Comput. Methods Funct. Theory 18 (2018), no. 4, 617–641. MR 3874885, DOI 10.1007/s40315-018-0243-5
Michel Hervé, Quelques propriétés des applications analytiques d’une boule à $m$ dimensions dan elle-même, J. Math. Pures Appl. (9) 42 (1963), 117–147 (French). MR 159962
Xiao Jun Huang, A non-degeneracy property of extremal mappings and iterates of holomorphic self-mappings, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 3, 399–419. MR 1310634
Marek Jarnicki and Peter Pflug, Invariant distances and metrics in complex analysis, De Gruyter Expositions in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1993. MR 1242120, DOI 10.1515/9783110870312
Anders Karlsson, Non-expanding maps and Busemann functions, Ergodic Theory Dynam. Systems 21 (2001), no. 5, 1447–1457. MR 1855841, DOI 10.1017/S0143385701001699
Anders Karlsson, On the dynamics of isometries, Geom. Topol. 9 (2005), 2359–2394. MR 2209375, DOI 10.2140/gt.2005.9.2359
Peter R. Mercer, Complex geodesics and iterates of holomorphic maps on convex domains in $\textbf {C}^n$, Trans. Amer. Math. Soc. 338 (1993), no. 1, 201–211. MR 1123457, DOI 10.1090/S0002-9947-1993-1123457-0
Nikolai Nikolov and Maria Trybuła, Gromov hyperbolicity of the Kobayashi metric on $\Bbb {C}$-convex domains, J. Math. Anal. Appl. 468 (2018), no. 2, 1164–1178. MR 3852573, DOI 10.1016/j.jmaa.2018.09.005
Nikolai Nikolov, Pascal J. Thomas, and Maria Trybuła, Gromov (non-)hyperbolicity of certain domains in $\Bbb C^n$, Forum Math. 28 (2016), no. 4, 783–794. MR 3518388, DOI 10.1515/forum-2014-0113
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
Peter Pflug and Włodzimierz Zwonek, Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains, Forum Math. 30 (2018), no. 1, 159–170. MR 3739333, DOI 10.1515/forum-2016-0217
H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Lecture Notes in Math., Vol. 185, Springer, Berlin-New York, 1971, pp. 125–137. MR 304694
Michael Spivak, A comprehensive introduction to differential geometry. Vol. II, M. Spivak, Brandeis Univ., Waltham, MA, 1970. Published by M. Spivak. MR 271845
Sergio Venturini, Pseudodistances and pseudometrics on real and complex manifolds, Ann. Mat. Pura Appl. (4) 154 (1989), 385–402. MR 1043081, DOI 10.1007/BF01790358
S. E. Warschawski, On differentiability at the boundary in conformal mapping, Proc. Amer. Math. Soc. 12 (1961), 614–620. MR 131524, DOI 10.1090/S0002-9939-1961-0131524-8
J. Wolff, Sur une généralisation d’un théorème de Schwarz, C. R. Acad. Sci. Paris 182 (1926), 918–920.
Andrew M. Zimmer, Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type, Math. Ann. 365 (2016), no. 3-4, 1425–1498. MR 3521096, DOI 10.1007/s00208-015-1278-9
Andrew M. Zimmer, Gromov hyperbolicity, the Kobayashi metric, and $\Bbb {C}$-convex sets, Trans. Amer. Math. Soc. 369 (2017), no. 12, 8437–8456. MR 3710631, DOI 10.1090/tran/6909
Andrew Zimmer, Subelliptic estimates from Gromov hyperbolicity, Adv. Math. 402 (2022), Paper No. 108334, 94. MR 4397690, DOI 10.1016/j.aim.2022.108334