Existence of constant mean curvature 2-Spheres in Riemannian 3-spheres |
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Authors: | Da Rong Cheng Xin Zhou |
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Institution: | 1. University of Waterloo, Waterloo, Ontario, Canada;2. Cornell University, Ithaca, New York, USA |
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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. |
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