Browsing by Author "Chen, Jingwen"

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    Allen-Cahn equation and degenerate minimal hypersurface
    (2024) Chen, Jingwen; Gaspar Marques Da Silva, Pedro Henrique
    In this short note, we present new observations and examples concerning the existence and rigidity of solutions to the Allen-Cahn equation with degenerate minimal hypersurfaces as their limit interfaces.
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    Mean curvature flow and low energy solutions of the parabolic Allen-Cahn equation on the Three-Sphere
    (2023) Chen, Jingwen; Silva Pedro Henrique, Gaspar Marques Da
    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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    Morse theory for the Allen-Cahn functional
    (2023) Chen, Jingwen; Gaspar Marques Da Silva, Pedro Henrique