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arXiv · 2012.13379

Existence of constant mean curvature 2-spheres in Riemannian 3-spheres

Da Rong Cheng·Xin Zhou
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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.

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Da Rong Cheng, Xin Zhou. 2020-12-24. Existence of constant mean curvature 2-spheres in Riemannian 3-spheres. https://arxiv.org/abs/2012.13379

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