Existence of constant mean curvature 2spheres in Riemannian 3spheres
Abstract
We prove the existence of branched immersed constant mean curvature 2spheres in an arbitrary Riemannian 3sphere for almost every prescribed mean curvature, and moreover for all prescribed mean curvatures when the 3sphere is positively curved. To achieve this, we develop a minmax scheme for a weighted Dirichlet energy functional. There are three main ingredients in our approach: a biharmonic approximation procedure to obtain compactness of the new functional, a derivative estimate of the minmax values to gain energy upper bounds for minmax 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.13379
 Bibcode:
 2020arXiv201213379C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Analysis of PDEs
 EPrint:
 55 pages. Acknowledgement section restored. No other changes from v2. To appear in Communications on Pure and Applied Mathematics