Abstract
We determine the asymptotic behavior of the admissible growth of the quasiconformality coefficient in a general global injectivity theorem for immersions of sub-Riemannian manifolds of conformally parabolic type. In the model case of a contact immersion of the Heisenberg group in itself, the asymptotic behavior of the admissible growth of the quasiconformality coefficient for which the mapping is still globally invertible was found by the author earlier.
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M. A. Lavrent’ev, “On a Differential Criterion for Homeomorphic Mappings of Three-Dimensional Domains,” Dokl. Akad. Nauk SSSR 20, 241–242 (1938).
V. A. Zorich, “A Theorem of M.A. Lavrent’ev on Quasiconformal Space Maps,” Mat. Sb. 74(3), 417–433 (1967) [Math. USSR, Sb. 3, 389–403 (1967)].
V. A. Zorich, “The Global Homeomorphism Theorem for Space Quasiconformal Mappings, Its Development and Related Open Problems,” in Quasiconformal Space Mappings (Springer, Berlin, 1992), Lect. Notes Math. 1508, pp. 132–148.
V. A. Zorich, “Quasi-conformal Maps and the Asymptotic Geometry of Manifolds,” Usp. Mat. Nauk 57(3), 3–28 (2002) [Russ. Math. Surv. 57, 437–462 (2002)].
V. A. Zorich, “Admissible Order of Growth of the Quasiconformality Characteristic in Lavrent’ev’s Theorem,” Dokl. Akad. Nauk SSSR 181(3), 530–533 (1968) [Sov. Math., Dokl. 9, 866–869 (1968)].
M. Gromov, “Hyperbolic Manifolds, Groups and Actions,” in Riemann Surfaces and Related Topics: Proc. Conf., Stony Brook, 1978 (Princeton Univ. Press, Princeton, NJ, 1981), Ann. Math. Stud. 97, pp. 183–213.
M. Gromov, Metric Structures for Riemannian and Non-Riemannian Spaces, with appendices by M. Katz, P. Pansu, and S. Semmes (Birkhäuser, Boston, 1999).
V. A. Zorich, “Quasiconformal Immersions of Riemannian Manifolds, and a Picard Type Theorem,” Funkts. Anal. Prilozh. 34(3), 37–48 (2000) [Funct. Anal. Appl. 34, 188–196 (2000)].
V. A. Zorich, “Asymptotics at Infinity of the Admissible Growth of the Quasi-conformality Coefficient and the Injectivity of Immersions of Riemannian Manifolds,” Usp. Mat. Nauk 58(3), 191–192 (2003) [Russ. Math. Surv. 58, 624–626 (2003)].
V. A. Zorich, “Asymptotics of the Admissible Growth of the Coefficient of Quasiconformality at Infinity and Injectivity of Immersions of Riemannian Manifolds,” Publ. Inst. Math., Nouv. Sér. 75, 53–57 (2004).
V. A. Zorich, “Asymptotic Geometry and Conformal Types of Carnot-Carathéodory Spaces,” Geom. Funct. Anal. 9(2), 393–411 (1999).
V. A. Zorich, “On Contact Quasi-conformal Immersions,” Usp. Mat. Nauk 60(2), 161–162 (2005) [Russ. Math. Surv. 60, 382–384 (2005)].
V. A. Zorich, “Contact Quasiconformal Immersions,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 253, 81–87 (2006) [Proc. Steklov Inst. Math. 253, 71–77 (2006)].
V. A. Zorich and V. M. Kessel’man, “On the Conformal Type of a Riemannian Manifold,” Funkts. Anal. Prilozh. 30(2), 40–55 (1996) [Funct. Anal. Appl. 30, 106–117 (1996)].
V. A. Zorich, “Asymptotic Behaviour at Infinity of the Admissible Growth of the Quasi-conformality Coefficient and the Invertibility of Contact Immersions of the Heisenberg Group,” Usp. Mat. Nauk 65(3), 191–192 (2010) [Russ. Math. Surv. 65, 591–592 (2010)].
V. G. Maz’ya, Sobolev Spaces (Leningrad Gos. Univ., Leningrad, 1985; Springer, Berlin, 1985).
A. Grigor’yan, “Isoperimetric Inequalities and Capacities on Riemannian Manifolds,” in The Maz’ya Anniversary Collection (Birkhäuser, Basel, 1999), Vol. 1, Oper. Theory: Adv. Appl. 109, pp. 139–153.
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Original Russian Text © V.A. Zorich, 2012, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2012, Vol. 279, pp. 81–85.
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Zorich, V.A. Asymptotic behavior at infinity of the admissible growth of the quasiconformality coefficient and the injectivity of immersions of sub-Riemannian manifolds. Proc. Steklov Inst. Math. 279, 73–77 (2012). https://doi.org/10.1134/S008154381208007X
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DOI: https://doi.org/10.1134/S008154381208007X