2023/07/19 by Costante Bellettini, Bellettini, Costante, Kobe Marshall-Stevens +1
Mathematics · Medicine · #35J20 #35J61 #35J93 #49Q20 #53A10 #53C21 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Pelvic and Acetabular Injuries
paper · pdf · doi:10.48550/arxiv.2307.10388
openalex publication_date 2023/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In compact Riemannian manifolds of dimension 3 or higher with positive Ricci curvature, we prove that every constant mean curvature hypersurface produced by the Allen-Cahn min-max procedure of Bellettini-Wickramasekera (with constant prescribing function) is a local minimiser of the natural area-type functional around each isolated singularity. In particular, every tangent cone at each isolated singularity of the resulting hypersurface is area-minimising. As a consequence, for any real λ we show, through a surgery procedure, that for a generic 8-dimensional compact Riemannian manifold with positive Ricci curvature there exists a closed embedded smooth hypersurface of constant mean curvature λ; the minimal case (λ = 0) of this result was obtained in work by Chodosh-Liokumovich-Spolaor.