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Existence of constant mean curvature 2-spheres in Riemannian 3-spheres
- Publication Year :
- 2020
-
Abstract
- We prove the existence of branched immersed constant mean curvature 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.<br />Comment: 55 pages. Acknowledgement section restored. No other changes from v2. To appear in Communications on Pure and Applied Mathematics
- Subjects :
- Mathematics - Differential Geometry
Mathematics - Analysis of PDEs
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2012.13379
- Document Type :
- Working Paper