Abstract
We review some recent results on existence and regularity of Monge-Ampère exhaustions on the smoothly bounded strongly pseudoconvex domains, which admit at least one such exhaustion of sufficiently high regularity. A main consequence of our results is the fact that the Kobayashi pseudo-metric \(\kappa \) on an appropriate open subset of each of the above domains is actually a smooth Finsler metric. The class of domains to which our result apply is very large. It includes for instance all smoothly bounded strongly pseudoconvex complete circular domains and all their sufficiently small deformations.
Dedicated to Kang-Tae Kim for his sixtieth birthday
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References
Abate, M., Patrizio, G.: Finsler Metrics - A Global Approach, Lecture Notes in Mathematics, vol. 1591. Springer-Verlag (1994)
Bland, J., Duchamp, T.: Moduli for pointed convex domains. Invent. Math. 104, 61–112 (1991)
Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann Finsler Geometry. Springer-Verlag (2000)
Blocki, Z.: The \(C^{1,1}\) regularity of the pluricomplex Green function. Michigan Math. J. 47, 211–215 (2000)
Chern, S.-S.: Finsler geometry is just Riemannian geometry without the quadratic restriction. Notices Amer. Math. Soc. 43, 959–963 (1996)
Demailly, J.-P.: Measures de Monge-Ampère et measures pluriharmonic. Math. Z. 194, 519–564 (1987)
Guan, B.: The Dirichlet problem for complex Monge-Ampère equations and regularity of the pluri-complex Green function. Comm. Anal. Geom. 6, 687–703 (1998)
Hill, C.D., Taylor, M.: Integrability of rough almost complex structures. J. Geom. Anal. 13(1), 163–172 (2003)
Jarnicki, M., Pflug, P.: Invariant Distances and Metrics in Complex Analysis. Walter de Gruyter GmbH & Co. KG, Berlin (2013)
Kobayashi, S.: Hyperbolic Manifolds and Holomorphic Mappings. Dekker, New York (1970)
Kobayashi, S.: Hyperbolic Complex Spaces. Springer, New York (1998)
Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109, 427–474 (1981)
Malgrange, B.: Sur l’integrabilite des structure presque-complex. Symposia Math, vol. II, 289–296. Academic Press (1969)
Newlander, A., Nirenberg, L.: Complex analytic coordinates in almost complex manifold. Ann. Math. 65, 391–404 (1957)
Nijenhuis, A., Woolf, W.: Some integration problem in almost-complex manifolds. Ann. Math. 77, 424–489 (1963)
Pang, M.-Y.: Smoothnes of the Kobayashi metric of nonconvex domains. Intern. J. Math. 4(6), 953–987 (1993)
Patrizio, G.: Parabolic exhaustions for strictly convex domains. Manuscripta Math. 47, 271–309 (1984)
Patrizio, G.: A characterization of complex manifolds biholomorphic to a circular domain. Math. Z. 189, 343–363 (1985)
Patrizio, G.: Disques extrémaux de Kobayashi et équation de Monge-Ampère complex. C. R. Acad. Sci. Paris, Série I, 305, 721–724 (1987)
Patrizio, G., Spiro, A.: Monge-Ampère equations and moduli spaces of manifolds of circular type. Adv. Math. 223, 174–197 (2010)
Patrizio, G., Spiro, A.: Foliations by stationary disks of almost complex domains. Bull. Sci. Math. 134, 215–234 (2010)
Patrizio, G., Spiro, A.: Modular data and regularity of Monge-Ampère exhaustions and of Kobayashi distance. Math. Ann. 362, 425–449 (2015)
Patrizio, G., Spiro, A.: Propagation of regularity for Monge-Ampre exhaustions and Kobayashi metrics (2017). arXiv:1707.09041
Spiro, A.: The structure equations of a complex Finsler manifold. Asian J. Math. 5, 291–326 (2001)
Webster, S.: A new proof of the Newlander-Nirenberg theorem. Math. Zeit. 201, 303–316 (1989)
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This research was partially supported by the Project MIUR “Real and Complex Manifolds: Geometry, Topology and Harmonic Analysis” and by GNSAGA of INdAM.
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Patrizio, G., Spiro, A. (2018). Regularity of Kobayashi Metric. In: Byun, J., Cho, H., Kim, S., Lee, KH., Park, JD. (eds) Geometric Complex Analysis. Springer Proceedings in Mathematics & Statistics, vol 246. Springer, Singapore. https://doi.org/10.1007/978-981-13-1672-2_24
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DOI: https://doi.org/10.1007/978-981-13-1672-2_24
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