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Harmonic flow of geometric structures

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Abstract

We give a twistorial interpretation of geometric structures on a Riemannian manifold, as sections of homogeneous fibre bundles, following an original insight by Wood (Differ Geom Appl 19:193–210, 2003). The natural Dirichlet energy induces an abstract harmonicity condition, which gives rise to a geometric gradient flow. We establish a number of analytic properties for this flow, such as uniqueness, smoothness, short-time existence, and some sufficient conditions for long-time existence. This description potentially subsumes a large class of geometric PDE problems from different contexts. As applications, we recover and unify a number of results in the literature: for the isometric flow of \(\text {G}_2\)-structures, by Grigorian (Adv Math 308:142–207, 2017; Calculas Variat Partial Differ Equ 58:157, 2019), Bagaglini (J Geom Anal, 2009), and Dwivedi-Gianniotis-Karigiannis (J Geom Anal 31(2):1855-1933, 2021); and for harmonic almost complex structures, by He (Energy minimizing harmonic almost complex structures, 2019) and He-Li (Trans Am Math Soc 374(9):6179–6199, 2021). Our theory also establishes original properties regarding harmonic flows of parallelisms and almost contact structures.

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Acknowledgements

The authors would like to thank the anonymous referee for numerous constructive suggestions to the original manuscript.

Funding

This project started during a visiting chair by EL, sponsored by Unicamp and Institut Français du Brésil, in 2017. The present article stems from an ongoing CAPES-COFECUB bilateral collaboration (2018-2021), Granted by Brazilian Coordination for the Improvement of Higher Education Personnel (CAPES) - Finance Code 001 [88881.143017/2017-01], and Campus France [MA 898/18]. HSE has been funded by São Paulo Research Foundation (Fapesp) [2017/20007-0] & [2018/21391-1] and the Brazilian National Council for Scientific and Technological Development (CNPq) [307217/2017-5]. Over its revision period, the project was further supported by the collaboration BRIDGES: Brazil-France interplays in Gauge Theory, extremal structures and stability, Granted by Fapesp [2021/04065-6] and the French National Research Agency [ANR-21-CE40-0017)].

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Authors and Affiliations

  1. LMBA, CNRS UMR 6205, University Brest, 29238, Brest, France

    Eric Loubeau

  2. University of Campinas (Unicamp), Campinas, Brazil

    Henrique N. Sá Earp

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  1. Eric Loubeau
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  2. Henrique N. Sá Earp
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Both authors contributed equally to the manuscript.

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Correspondence to Henrique N. Sá Earp.

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Loubeau, E., Sá Earp, H.N. Harmonic flow of geometric structures. Ann Glob Anal Geom 64, 23 (2023). https://doi.org/10.1007/s10455-023-09928-7

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