Existence of constant mean curvature 2‐Spheres in Riemannian 3‐spheres

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.