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Regularity of Kobayashi Metric

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Geometric Complex Analysis

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 246))

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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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Acknowledgements

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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Authors and Affiliations

  1. Dipartimento Matematica e Informatica “U. Dini”, Università di Firenze, Florence, Italy

    Giorgio Patrizio

  2. Istituto Nazionale di Alta Matematica “Francesco Severi”, Rome, Italy

    Giorgio Patrizio

  3. Scuola di Scienze e Tecnologie, Università di Camerino, Via Madonna delle Carceri, 62032, Camerino (Macerata), Italy

    Andrea Spiro

Authors
  1. Giorgio Patrizio
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  2. Andrea Spiro
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Corresponding author

Correspondence to Andrea Spiro .

Editor information

Editors and Affiliations

  1. Department of Mathematics Education, Kyungnam University, Changwon, Korea (Republic of)

    Jisoo Byun

  2. Department of Mathematics, Pusan National University, Busan, Korea (Republic of)

    Hong Rae Cho

  3. Center for Mathematical Challenges, Korea Institute for Advanced Study, Seoul, Korea (Republic of)

    Sung Yeon Kim

  4. Department of Mathematics, Gyeongsang National University, Jinju, Korea (Republic of)

    Kang-Hyurk Lee

  5. Department of Mathematics, Kyung Hee University, Seoul, Korea (Republic of)

    Jong-Do Park

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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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