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Mean Curvature Flow and Low Energy Solutions of the Parabolic Allen-Cahn Equation on the Three-Sphere

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Abstract

In this article, we study eternal solutions to the Allen-Cahn equation in the 3-sphere, in view of the connection between the gradient flow of the associated energy functional, and the mean curvature flow. We construct eternal integral Brakke flows that connect Clifford tori to equatorial spheres, and study a family of such flows, in particular their symmetry properties. Our approach is based on the realization of Brakke’s motion by mean curvature as a singular limit of Allen-Cahn gradient flows, as studied by Ilmanen (J Differ Geom 38(2):417–461, 1993) and Tonegawa (Hiroshima Math J 33(3): 323–341, 2003), and it uses the classification of ancient gradient flows in spheres, by Choi and Mantoulidis (Amer J Math, 2019), as well as the rigidity of stationary solutions with low Morse index proved by Hiesmayr (arXiv:2007.08701 [math.DG], 2020).

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Acknowledgements

We would like to thank André Neves for his support, and for many invaluable discussions and suggestions. PG was partially supported by Prof. Neves’ Simons Investigator Award.

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Authors and Affiliations

  1. Department of Mathematics, The University of Chicago, 5734 S University Ave, Chicago, IL, 60615, USA

    Jingwen Chen & Pedro Gaspar

  2. Facultad de Matemáticas, Pontifica Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

    Pedro Gaspar

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  1. Jingwen Chen
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Correspondence to Pedro Gaspar.

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Chen, J., Gaspar, P. Mean Curvature Flow and Low Energy Solutions of the Parabolic Allen-Cahn Equation on the Three-Sphere. J Geom Anal 33, 283 (2023). https://doi.org/10.1007/s12220-023-01347-1

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