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Vaisman manifolds and transversally Kähler–Einstein metrics

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Abstract

We use the transverse Kähler–Ricci flow on the canonical foliation of a closed Vaisman manifold to deform the Vaisman metric into another Vaisman metric with a transverse Kähler–Einstein structure. We also study the main features of such a manifold. Among other results, using techniques from the theory of parabolic equations, we obtain a direct proof for the short-time existence of the solution for transverse Kähler–Ricci flow on Vaisman manifolds, recovering in a particular setting a result of Bedulli et al. (J Geom Anal 28:697–725, 2018), but without employing the Molino structure theorem. Moreover, we investigate Einstein–Weyl structures in the setting of Vaisman manifolds and find their relationship with quasi-Einstein metrics. Some examples are also provided to illustrate the main results.

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Acknowledgements

The authors would like to thank Professor Luigi Vezzoni for bringing the article [3] to their attention and for useful comments on the first draft of this paper. They are also extremely grateful to Professor Liviu Ornea for the useful suggestions and remarks that led to the improvement of the article. The authors thank the referee for very carefully reading a first version of the paper and for the most useful suggestions.

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Authors and Affiliations

  1. Faculty of Applied Sciences, University “Politehnica” of Bucharest, 313 Splaiul Independenţei, 060042, Bucharest, Romania

    Vladimir Slesar & Gabriel-Eduard Vîlcu

  2. Faculty of Mathematics and Informatics, Research Center in Geometry, Topology and Algebra, University of Bucharest, 14 Academiei str., 70109, Bucharest, Romania

    Vladimir Slesar & Gabriel-Eduard Vîlcu

  3. “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie, Nr. 13, 050711, Bucharest, Romania

    Gabriel-Eduard Vîlcu

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  1. Vladimir Slesar
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Correspondence to Vladimir Slesar.

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Partially supported by Romanian Ministry of Education and Research, Program PN-III, Project number PN-III-P4-ID-PCE-2020-0025, Contract 30/04.02.2021.

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Slesar, V., Vîlcu, GE. Vaisman manifolds and transversally Kähler–Einstein metrics. Annali di Matematica 202, 1855–1876 (2023). https://doi.org/10.1007/s10231-023-01304-3

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