Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T06:20:14.026Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE BOUNDARY BEHAVIOUR OF FRIDMAN INVARIANTS

Part of: Holomorphic mappings and correspondences Geometric convexity

Published online by Cambridge University Press:  22 September 2021

SHICHAO YANG
SHICHAO YANG*
Affiliation:
School of Mathematical Sciences, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai, 200240, PR China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the Fridman invariant defined using the Carathéodory pseudodistance does not always go to 1 near strongly Levi pseudoconvex boundary points and it always goes to 0 near nonpseudoconvex boundary points. We also discuss whether Fridman invariants can be extended continuously to some boundary points of domains constructed by deleting compact subsets from other domains.

Keywords

Fridman invariantstrongly pseudoconvexnonpseudoconvex

MSC classification

Primary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
Secondary: 32F45: Invariant metrics and pseudodistances
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 3 , June 2022 , pp. 482 - 489
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bharali, G., ‘A new family of holomorphic homogeneous regular domains and some questions on the squeezing function’, Preprint, 2021, arXiv:2103.09227.Google Scholar
Deng, F., Guan, Q. and Zhang, L., ‘Some properties of squeezing functions on bounded domains’, Pacific J. Math. 257 (2012), 319–341.CrossRefGoogle Scholar
Deng, F., Guan, Q. and Zhang, L., ‘Properties of squeezing functions and global transformations of bounded domains’, Trans. Amer. Math. Soc. 368 (2016), 2679–2696.CrossRefGoogle Scholar
Deng, F., Wang, Z., Zhang, L. and Zhou, X., ‘Holomorphic invariants of bounded domains’, J. Geom. Anal. 30 (2020), 1204–1217.CrossRefGoogle Scholar
Fornaess, J. E. and Wold, E. F., ‘An estimate for the squeezing function and estimates of invariant metrics’, in: Complex Analysis and Geometry, Springer Proceedings in Mathematics and Statistics, 144 (eds. F. Bracci, J. Byun, H. Gaussier, K. Hirachi, K.T. Kim and N. Shcherbina) (Springer, Tokyo, 2015), 135–147.CrossRefGoogle Scholar
Fridman, B. L., ‘On the imbedding of a strictly pseudoconvex domain in a polyhedron’, Dokl. Akad. Nauk SSSR 249 (1979), 63–67 (in Russian); Soviet Math. Dokl. 20 (1979), 1228–1232 (English translation).Google Scholar
Fridman, B. L., ‘Biholomorphic invariants of a hyperbolic manifold and some applications’, Trans. Amer. Math. Soc. 276 (1983), 685–698.Google Scholar
Jarnicki, M. and Pflug, P., Invariant Distances and Metrics in Complex Analysis (Walter de Gruyter, Berlin, 2013).CrossRefGoogle Scholar
Kim, K. T. and Zhang, L., ‘On the uniform squeezing property of bounded convex domains in ${\mathbb{C}}^n$ ’, Pacific J. Math. 282 (2016), 341–358.CrossRefGoogle Scholar
Krantz, S. G., Function Theory of Several Complex Variables (AMS Chelsea Publishing, Providence, RI, 1992).Google Scholar
Mahajan, P. and Verma, K., ‘A comparison of two biholomorphic invariants’, Internat. J. Math. 30 (2019), Article no. 1950012, 16 pages.CrossRefGoogle Scholar
Ng, T. W., Tang, C. C. and Tsai, J., ‘Fridman function, injectivity radius function and squeezing function’, Preprint, 2021, arXiv:2012.13159.CrossRefGoogle Scholar
Nikolov, N. and Trybula, M., ‘Estimates for the squeezing function near strictly pseudoconvex boundary points with applications’, J. Geom. Anal. 30 (2020), 1359–1365.CrossRefGoogle Scholar
Nikolov, N. and Verma, K., ‘On the squeezing function and Fridman invariants’, J. Geom. Anal. 30 (2020), 1218–1225.CrossRefGoogle Scholar
Rong, F. and Yang, S., ‘On the comparison of the Fridman invariant and the squeezing function’, Complex Var. Elliptic Equ., to appear, https://doi.org/10.1080/17476933.2020.1851210.CrossRefGoogle Scholar
Rong, F. and Yang, S., ‘On Fridman invariants and generalized squeezing functions’, Preprint, 2019.Google Scholar
Rong, F. and Yang, S., ‘On the generalized squeezing functions and Fridman invariants of special domains’, Preprint, 2020.Google Scholar