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Positive Hermitian curvature flow on nilpotent and almost-abelian complex Lie groups

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Abstract

We study the positive Hermitian curvature flow on the space of left-invariant metrics on complex Lie groups. We show that in the nilpotent case, the flow exists for all positive times and subconverges in the Cheeger–Gromov sense to a soliton. We also show convergence to a soliton when the complex Lie group is almost abelian. That is, when its Lie algebra admits a (complex) co-dimension one abelian ideal. Finally, we study solitons in the almost-abelian setting. We prove uniqueness and completely classify all left-invariant, almost-abelian solitons, giving a method to construct examples in arbitrary dimensions, many of which admit co-compact lattices.

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Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

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Acknowledgements

I am grateful to my advisor Ramiro Lafuente for his invaluable input and guidance. I also wish to thank Artem Pulemotov and Romina Arroyo for helpful discussions.

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  1. School of Mathematics and Physics, The University of Queensland, St Lucia, QLD, 4072, Australia

    James Stanfield

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  1. James Stanfield
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Correspondence to James Stanfield.

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This work was supported by an Australian Government Research Training Program (RTP) Scholarship.

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Stanfield, J. Positive Hermitian curvature flow on nilpotent and almost-abelian complex Lie groups. Ann Glob Anal Geom 60, 401–429 (2021). https://doi.org/10.1007/s10455-021-09782-5

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