Symmetric mean curvature flow on the n-sphere

Article Mathematics

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

Jingwen Chen et al.

Summary: In this article, the eternal solutions to the Allen-Cahn equation in the 3-sphere are studied, focusing on the relationship between the gradient flow of the associated energy functional and the mean curvature flow. Eternal integral Brakke flows connecting Clifford tori to equatorial spheres are constructed and the symmetry properties of such flows are analyzed. The 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 utilizes 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).

JOURNAL OF GEOMETRIC ANALYSIS (2023)

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Article Mathematics, Applied

Ground states of semilinear elliptic problems with applications to the Allen-Cahn equation on the sphere

Rayssa Caju et al.

Summary: This study focuses on solutions of nonlinear elliptic partial differential equations concerning ground state solutions, analyzing the properties and symmetry of ground state solutions under different spatial and boundary conditions to prove relevant conclusions.

CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS (2022)

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Article Mathematics