2020/07/17 by Fritz Hiesmayr, Hiesmayr, Fritz
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2007.08701
openalex publication_date 2020/07/17 · openalex created_date 2020/07/23 · openalex updated_date 2026/07/28
We prove a rigidity theorem in the style of Urbano for the Allen-Cahn equation on the three-sphere: the critical points with Morse index five are symmetric functions that vanish on a Clifford torus. Moreover they realise the fifth width of the min-max spectrum for the Allen-Cahn functional. We approach this problem by analysing the nullity and symmetries of these critical points. We then prove a suitable Frankel-type theorem for their nodal sets, generally valid in manifolds with positive Ricci curvature. This plays a key role in establishing the conclusion, and further allows us to derive ancillary rigidity results in spheres with larger dimension.