2023/06/28 by Da Rong Cheng, Xin Zhou · 2 citations
Mathematics · #Constant (computer programming) #Constant-mean-curvature surface #Curvature #Geometric Analysis and Curvature Flows #Geometry #Mathematical Approximation and Integration #Mathematical analysis #Mathematics #Mean curvature #Mean curvature flow #Nonlinear Partial Differential Equations #Physics #SPHERES #Upper and lower bounds
paper · doi:10.1002/cpa.22114
published in Communications on Pure and Applied Mathematics 76(11), 3374-3436 (Wiley)
openalex created_date 2022/07/25 · openalex publication_date 2023/06/28 · openalex updated_date 2026/05/21
Abstract We prove the existence of branched immersed constant mean curvature (CMC) 2‐spheres in an arbitrary Riemannian 3‐sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3‐sphere is positively curved. To achieve this, we develop a min‐max scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a bi‐harmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the min‐max values to gain energy upper bounds for min‐max sequences for almost every choice of mean curvature, and a Morse index estimate to obtain another uniform energy bound required to reach the remaining constant mean curvatures in the presence of positive curvature.