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Geometry and Topology of the Space of Plurisubharmonic Functions

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Abstract

Let \(\Omega \) be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in \(\Omega \). We study the metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application, we study the existence of local Kähler–Einstein metrics.

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Acknowledgements

The author is grateful for his supervisors Vincent Guedj and Said Asserda, for their support, suggestions and encouragement. The author wants to thank Ahmed Zeriahi for his very useful discussions and suggestions. The author would also like to thank Tat Dat Tô and Zakarias Sjöström Dyrefelt for their very careful reading of the preliminary version of this paper and their very useful discussions. This work has been finalized while the author was visiting the “Institut de Mathématiques de Toulouse” in march 2017 and he would like to thank them for their hospitality.

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  1. Department of Mathematics, Faculty of Sciences, Ibn Tofail University, PO 242, Kenitra, Morocco

    Soufian Abja

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  1. Soufian Abja
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Abja, S. Geometry and Topology of the Space of Plurisubharmonic Functions. J Geom Anal 29, 510–541 (2019). https://doi.org/10.1007/s12220-018-0009-3

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  • DOI: https://doi.org/10.1007/s12220-018-0009-3

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