A Kobayashi pseudo-distance for holomorphic bracket generating distributions
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Abstract:
We generalize the Kobayashi pseudo-distance to complex manifolds which admit holomorphic bracket generating distributions. The generalization is based on Chow’s theorem in sub-Riemannian geometry. Let $G$ be a linear semisimple Lie group. For a complex $G$-homogeneous manifold $M$ with a $G$-invariant holomorphic bracket generating distribution $D$, we prove that $(M,D)$ is Kobayashi hyperbolic if and only if the universal covering of $M$ is a canonical flag domain and the induced distribution is the superhorizontal distribution.References
- 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
- Dmitri N. Akhiezer, Lie group actions in complex analysis, Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, Braunschweig, 1995. MR 1334091, DOI 10.1007/978-3-322-80267-5
- Salomon Bochner and Deane Montgomery, Groups of differentiable and real or complex analytic transformations, Ann. of Math. (2) 46 (1945), 685–694. MR 14102, DOI 10.2307/1969204
- Francis E. Burstall and John H. Rawnsley, Twistor theory for Riemannian symmetric spaces, Lecture Notes in Mathematics, vol. 1424, Springer-Verlag, Berlin, 1990. With applications to harmonic maps of Riemann surfaces. MR 1059054, DOI 10.1007/BFb0095561
- James Carlson, Stefan Müller-Stach, and Chris Peters, Period mappings and period domains, Cambridge Studies in Advanced Mathematics, vol. 85, Cambridge University Press, Cambridge, 2003. MR 2012297
- Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung, Math. Ann. 117 (1939), 98–105 (German). MR 1880, DOI 10.1007/BF01450011
- D. van Dantzig and B. L. van der Waerden, Über metrisch homogene räume, Abh. Math. Sem. Univ. Hamburg 6 (1928), no. 1, 367–376 (German). MR 3069509, DOI 10.1007/BF02940622
- Jean-Pierre Demailly, Kobayashi pseudo-metrics, entire curves and hyperbolicity of algebraic varieties, Lecture Notes of a 7 hour series of courses given at the Grenoble Summer School in Mathematics, June 18-22, 2012 (94 pages).
- Franc Forstnerič, Hyperbolic complex contact structures on $\Bbb C^{2n+1}$, J. Geom. Anal. 27 (2017), no. 4, 3166–3175. MR 3708010, DOI 10.1007/s12220-017-9800-9
- Gregor Fels, Alan Huckleberry, and Joseph A. Wolf, Cycle spaces of flag domains, Progress in Mathematics, vol. 245, Birkhäuser Boston, Inc., Boston, MA, 2006. A complex geometric viewpoint. MR 2188135, DOI 10.1007/0-8176-4479-2
- È. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, Structure of Lie groups and Lie algebras, Current problems in mathematics. Fundamental directions, Vol. 41 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 5–259 (Russian). MR 1056486, DOI 10.1007/978-1-4614-1068-3_{2}
- Phillip Griffiths and Wilfried Schmid, Locally homogeneous complex manifolds, Acta Math. 123 (1969), 253–302. MR 259958, DOI 10.1007/BF02392390
- Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. MR 0277770
- Richard Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002. MR 1867362, DOI 10.1090/surv/091
- Kazufumi Nakajima, Homogeneous hyperbolic manifolds and homogeneous Siegel domains, J. Math. Kyoto Univ. 25 (1985), no. 2, 269–291. MR 794987, DOI 10.1215/kjm/1250521109
- 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, 1971, pp. 125–137. MR 0304694
- Joseph A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. MR 251246, DOI 10.1090/S0002-9904-1969-12359-1
- Shing Tung Yau, A general Schwarz lemma for Kähler manifolds, Amer. J. Math. 100 (1978), no. 1, 197–203. MR 486659, DOI 10.2307/2373880
Additional Information
- Aeryeong Seo
- Affiliation: School of Mathematics, Korea Institute for Advanced Study (KIAS), 85 Hoegiro (Cheongnyangni-dong 207-43), Dongdaemun-gu, Seoul 130-722, Republic of Korea
- MR Author ID: 984919
- Email: aeryeongseo@kias.re.kr
- Received by editor(s): January 17, 2017
- Received by editor(s) in revised form: August 17, 2017, September 12, 2017, November 18, 2017, and January 7, 2018
- Published electronically: September 18, 2018
- Additional Notes: This research was partially supported by the “Overseas Research Program for Young Scientists” through the Korea Institute for Advanced Study (KIAS)
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5023-5038
- MSC (2010): Primary 32Q45, 32M10; Secondary 53C17, 32F45
- DOI: https://doi.org/10.1090/tran/7537
- MathSciNet review: 3934476