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Invariant Metrics on the Complex Ellipsoid

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Abstract

We provide a class of geometric convex domains on which the Carathéodory–Reiffen metric, the Bergman metric, the complete Kähler–Einstein metric of negative scalar curvature are uniformly equivalent, but not proportional to each other. In a two-dimensional case, we provide a full description of curvature tensors of the Bergman metric on the weakly pseudoconvex boundary point and show that invariant metrics are proportional to each other if and only if the geometric convex domain is the Euclidean ball.

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Acknowledgements

This work was partially supported by NSF Grant DMS-1611745. I would like to thank my advisor Professor Damin Wu for many insights and deep encouragements. Also, I appreciate Prof. Yunhui Wu who pointed out the alternative proof of the Theorem 1 after submitting the paper on ArXiv and referees for carefully reading the paper and providing helpful comments on improving the paper. I appreciate to Hyun Chul Jang who is the Ph.D. candidate in Uconn for helpful discussions.

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  1. Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs, CT, 06269-3009, USA

    Gunhee Cho

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  1. Gunhee Cho
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Correspondence to Gunhee Cho.

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Cho, G. Invariant Metrics on the Complex Ellipsoid. J Geom Anal 31, 2088–2104 (2021). https://doi.org/10.1007/s12220-019-00333-w

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