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SPHERICALIZATION AND FLATTENING PRESERVE UNIFORM DOMAINS IN NONLOCALLY COMPACT METRIC SPACES

Part of: Geometric function theory Riemann surfaces

Published online by Cambridge University Press:  27 February 2020

YAXIANG LI ,
SAMINATHAN PONNUSAMY Open the ORCID record for SAMINATHAN PONNUSAMY [Opens in a new window]  and
QINGSHAN ZHOU
YAXIANG LI
Affiliation:
Department of Mathematics, Hunan First Normal University, Changsha, Hunan410205, PR China e-mail: yaxiangli@163.com
SAMINATHAN PONNUSAMY
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India e-mail: samy@iitm.ac.in
QINGSHAN ZHOU
Affiliation:
School of Mathematics and Big Data, Foshan University, Foshan, Guangdong528000, PR China e-mail: q476308142@qq.com
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Abstract

The main aim of this paper is to investigate the invariant properties of uniform domains under flattening and sphericalization in nonlocally compact complete metric spaces. Moreover, we show that quasi-Möbius maps preserve uniform domains in nonlocally compact spaces as well.

Keywords

uniform domainsflatteningsphericalizationquasi-Möbiusquasisymmetricquasihyperbolic metric

MSC classification

Primary: 30C65: Quasiconformal mappings in $^n$, other generalizations 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions)
Secondary: 30C20: Conformal mappings of special domains
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 1 , February 2022 , pp. 68 - 89
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by W. Moors

The first author was supported by NNSF of China (nos 11601529, 11671127, 11971124). The third author was supported by NNSF of China (nos 11901090, 11571216), and by Department of Education of Guangdong Province, China (grant nos 2018KQNCX285 and 2018KTSCX245).

References

Balogh, Z. and Buckley, S., ‘Sphericalization and flattening’, Conf. Geom. Dyn. 9 (2005), 76–101.CrossRefGoogle Scholar
Bonk, M., Heinonen, J. and Koskela, P., ‘Uniformizing Gromov hyperbolic domains’, Asterisque 270 (2001), 1–99.Google Scholar
Bonk, M. and Kleiner, B., ‘Rigidity for quasi-Möbius group actions’, J. Differential Geom. 61 (2002), 81–106.Google Scholar
Buckley, S. M. and Herron, D., ‘Uniform spaces and weak slice spaces’, Conf. Geom. Dyn. 11 (2007), 191–206.CrossRefGoogle Scholar
Buckley, S. M., Herron, D. and Xie, X., ‘Metric space inversions, quasihyperbolic distance, and uniform spaces’, Indiana Univ. Math. J. 57 (2008), 837–890.Google Scholar
Capogna, L., Garofalo, N. and Nhieu, D.-M., ‘Examples of uniform and NTA domains in Carnot groups’, in: Proceedings on Analysis and Geometry, Novosibirsk, 1999, Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat. (Sobolev Institute Press, Novosibirsk, 2000), 103–121.Google Scholar
Capogna, L. and Tang, P., ‘Uniform domains and quasiconformal mappings on the Heisenberg group’, Manuscripta Math. 86(3) (1995), 267–281.CrossRefGoogle Scholar
Durand-Cartagena, E. and Li, X., ‘Preservation of p-Poincaré inequality for large p under sphericalization and flattening’, Illinois J. Math. 59(4) (2015), 1043–1069.CrossRefGoogle Scholar
Durand-Cartagena, E. and Li, X., ‘Preservation of bounded geometry under sphericalization and flattening: quasiconvexity and ∞-Poincaré inequality’, Ann. Acad. Sci. Fenn. Math. 42(1) (2017), 303–324.CrossRefGoogle Scholar
Franchi, B., Penso, V. and Serapioni, R., ‘Remarks on Lipschitz domains in Carnot groups’, in: Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Series, 5 (eds. Stefani, G., Boscain, U., Gauthier, J. P., Sarychev, A. and Sigalotti, M.) (Springer, Cham, 2014).Google Scholar
Gehring, F. W., ‘Uniform domains and the ubiquitous quasidisks’, Jahresber. Dtsch. Math.-Ver. 89 (1987), 88–103.Google Scholar
Gehring, F. W. and Hag, K., The ubiquitous quasidisk, Mathematical Surveys and Monographs, 184 (American Mathematical Society, Providence, RI, 2012), with contributions by Ole Jacob Broch.CrossRefGoogle Scholar
Gehring, F. W. and Osgood, B. G., ‘Uniform domains and the quasi-hyperbolic metric’, J. Anal. Math. 36 (1979), 50–74.CrossRefGoogle Scholar
Greshnov, A. V., ‘On uniform and NTA-domains on Carnot groups’, Sibirsk. Mat. Zh. 42 (2001), 1018–1035.Google Scholar
Herron, D., Shanmugalingam, N. and Xie, X., ‘Uniformity from Gromov hyperbolicity’, Illinois J. Math. 52 (2008), 1065–1109.CrossRefGoogle Scholar
Huang, M., Rasila, A., Wang, X. and Zhou, Q., ‘Semisolidity and locally weak quasisymmetry of homeomorphisms in metric spaces’, Stud. Math. 242 (2018), 267–301.CrossRefGoogle Scholar
John, F., ‘Rotation and strain’, Commun. Pure Appl. Math. 14 (1961), 391–413.CrossRefGoogle Scholar
Li, X. and Shanmugalingam, N., ‘Preservation of bounded geometry under sphericalization and flattening’, Indiana Math. J. 64(5) (2015), 1303–1341.CrossRefGoogle Scholar
Li, Y., Vuorinen, M. and Zhou, Q., ‘Weakly quasisymmetric maps and uniform spaces’, Comput. Meth. Funct. Theor. 18(4) (2018), 689–715.CrossRefGoogle Scholar
Martio, O., ‘Definitions of uniform domains’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 5 (1980), 197–205.CrossRefGoogle Scholar
Martio, O. and Sarvas, J., ‘Injectivity theorems in plane and space’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 4 (1979), 383–401.CrossRefGoogle Scholar
Tukia, P. and Väisälä, J., ‘Quasisymmetric embeddings of metric spaces’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 5 (1980), 97–114.CrossRefGoogle Scholar
Väisälä, J., ‘Quasi-Möbius maps’, J. Anal. Math. 44 (1984/85), 218–234.CrossRefGoogle Scholar
Väisälä, J., ‘Free quasiconformality in Banach spaces. I’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 15 (1990), 355–379.CrossRefGoogle Scholar
Väisälä, J., ‘Free quasiconformality in Banach spaces. II’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 16 (1991), 255–310.CrossRefGoogle Scholar
Väisälä, J., ‘Relatively and inner uniform domains’, Conf. Geom. Dyn. 2 (1998), 56–88.CrossRefGoogle Scholar
Väisälä, J., ‘The free quasiworld, freely quasiconformal and related maps in Banach spaces’, in: Quasiconformal Geometry and Dynamics, Banach Center Publications, 48 (Polish Academy of Sciences, Warszawa, 1999), 55–118.Google Scholar
Väisälä, J., ‘Gromov hyperbolic spaces’, Expo. Math. 23 (2005), 187–231.CrossRefGoogle Scholar
Väisälä, J., ‘Hyperbolic and uniform domains in Banach spaces’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 30 (2005), 261–302.Google Scholar
Xie, X., ‘Quasimöbius maps preserve uniform domains’, Ann. Acad. Sci. Fenn. Ser. A. I. Math. 32 (2007), 481–495.Google Scholar
Zhou, Q., Li, Y. and Li, X., ‘Sphericalization and flattening with their applications in quasimetric measure spaces’, Preprint, arXiv:1911.01760v1.Google Scholar