Comptes Rendus
Complex analysis and geometry
Topological invariants and Holomorphic Mappings
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 829-844.

Invariants for Riemann surfaces covered by the disc and for hyperbolic manifolds in general involving minimizing the measure of the image over the homotopy and homology classes of closed curves and maps of the k-sphere into the manifold are investigated. The invariants are monotonic under holomorphic mappings and strictly monotonic under certain circumstances. Applications to holomorphic maps of annular regions in and tubular neighborhoods of compact totally real submanifolds in general in n , n2, are given. The contractibility of a hyperbolic domain with contracting holomorphic mapping is explained.

Published online:
DOI: 10.5802/crmath.336

Robert E. Greene 1; Kang-Tae Kim 2; Nikolay V. Shcherbina 3

1 Department of Mathematics, University of California, Los Angeles, CA 90095 U.S.A.
2 Department of Mathematics, Pohang University of Science and Technology, Pohang City 37673 South Korea
3 Department of Mathematics, University of Wuppertal, 42119 Wuppertal, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Robert E. Greene; Kang-Tae Kim; Nikolay V. Shcherbina. Topological invariants and Holomorphic Mappings. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 829-844. doi : 10.5802/crmath.336.

[1] Lars V. Ahlfors Complex analysis. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics, McGraw-Hill, 1979 | Zbl

[2] Theodore J. Barth The Kobayashi distance induces the standard topology, Proc. Am. Math. Soc., Volume 35 (1972), pp. 439-441 | DOI | MR | Zbl

[3] Clifford J. Earle; Richard S. Hamilton A fixed point theorem for holomorphic mappings, Global Analysis (Proceedings of Symposia in Pure Mathematics), Volume 16, American Mathematical Society, 1968, pp. 61-65 | DOI

[4] Theodore W. Gamelin; Robert E. Greene Introduction to topology, Dover Publications, 1999, xiv+234 pages

[5] Gennadiĭ M. Goluzin Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, 26, American Mathematical Society, 1969, vi+676 pages | DOI

[6] Ian Graham Boundary behavior of the Carathéodory, Kobayashi, and Bergman metrics on strongly pseudoconvex domains in C n with smooth boundary, Trans. Am. Math. Soc., Volume 207 (1975), pp. 219-240 | MR | Zbl

[7] Robert E. Greene; Kang-Tae Kim; Steven G. Krantz The geometry of complex domains, Progress in Mathematics, 291, Birkhäuser, 2011, xiv+303 pages | DOI

[8] Robert E. Greene; Steven G. Krantz Function theory of one complex variable, Graduate Studies in Mathematics, 40, American Mathematical Society, 2006, xix+504 pages

[9] Robert E. Greene; Hung-Hsi Wu 𝒞 approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. Éc. Norm. Supér., Volume 12 (1979) no. 1, pp. 47-84 | DOI | MR | Numdam

[10] Herbert Grötzsch Zur Theorie der Verschiebung bei schlichter konformer Abbildung, Comment. Math. Helv., Volume 8 (1935) no. 1, pp. 382-390 | DOI | MR | Zbl

[11] Allen Hatcher Algebraic topology, Cambridge University Press, 2002, xii+544 pages

[12] Michel Hervé Several complex variables. Local theory, Tata Institute of Fundamental Research Studies in Mathematics, 1, Oxford University Press, 1963 | Numdam

[13] Kang-Tae Kim; Hanjin Lee Schwarz’s lemma from a differential geometric viewpoint, IISc Lecture Notes Series, 2, World Scientific, 2011, xvi+82 pages

[14] Shoshichi Kobayashi Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, 2, Marcel Dekker, 1970, ix+148 pages

[15] Shoshichi Kobayashi Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer, 1998 | DOI

[16] John E. Littlewood Quelques consequences de l’hypothese que la fonction ζ(s) de Riemann n’a pas de zeros dans le demi-plan (s)>1/2, C. R. Acad. Sci. Paris, Volume 154 (1912), pp. 263-266

[17] John W. Milnor Morse theory. Based on lecture notes by M. Spivak and R. Wells, Annals of Mathematics Studies, 51, Princeton University Press, 1963, vi+153 pages

[18] Marston Morse The critical points of a function of n variables, Trans. Am. Math. Soc., Volume 33 (1931), pp. 72-91 | MR | Zbl

[19] Tommaso Pacini Extremal length in higher dimensions and complex systolic inequalities, J. Geom. Anal., Volume 31 (2021) no. 5, pp. 5073-5093 | DOI | MR | Zbl

[20] Peter Petersen Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, 2006, xvi+401 pages | MR | Numdam

[21] Rene de Possel Sur quelques propriétés de la représentation conforme des domaines multiplement connexes, en relation avec le théorème des fentes parallèles, Math. Ann., Volume 107 (1933) no. 1, pp. 496-504 | DOI | Zbl

[22] Hans-Jörg Reiffen Die Carathéodorysche Distanz und ihre zugehörige Differentialmetrik, Math. Ann., Volume 161 (1965), pp. 315-324 | DOI | Zbl

[23] H. L. Royden Remarks on the Kobayashi metric, Lecture Notes in Mathematics, 185, Springer, 1970, pp. 125-137

[24] H. L. Royden Report on the Teichmüller metric, Proc. Natl. Acad. Sci. USA, Volume 65 (1970), pp. 497-499 | DOI | MR | Zbl

[25] Walter Rudin Real and complex analysis, McGraw-Hill, 1987

[26] John H. C. Whitehead Combinatorial homotopy II, Bull. Am. Math. Soc., Volume 55 (1949), pp. 453-496 | DOI | MR | Zbl

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