[Paper Review] Existence of constant mean curvature 2-spheres in Riemannian 3-spheres

Da Rong Cheng, Xin Zhou|arXiv (Cornell University)|Dec 24, 2020

Geometric Analysis and Curvature Flows50 references4 citations

TL;DR

This paper establishes the existence of branched immersed constant mean curvature (CMC) 2-spheres in any Riemannian 3-sphere for almost every prescribed mean curvature H > 0, and for all H > 0 when the 3-sphere has positive Ricci curvature. The authors develop a novel min-max scheme for a weighted Dirichlet energy functional, leveraging bi-harmonic approximation, derivative estimates of min-max values, and Morse index bounds to achieve compactness and energy control.

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.

Motivation & Objective

• To establish a general existence theory for closed, branched immersed CMC 2-spheres in arbitrary Riemannian 3-spheres with controlled topology.
• To resolve the open problem of constructing CMC 2-spheres for intermediate mean curvatures H > 0 beyond the known cases H = 0 and very large H.
• To confirm a conjecture by Rosenberg-Smith on the existence of branched immersed CMC 2-spheres for all H > 0 in positively curved 3-spheres.
• To extend the Sacks-Uhlenbeck min-max theory for minimal surfaces to the CMC setting in 3-spheres.

Proposed method

• Develop a min-max scheme for a weighted Dirichlet energy functional to construct CMC surfaces via variational methods.
• Implement a bi-harmonic approximation procedure to ensure compactness of the min-max sequences under the new functional.
• Derive a derivative estimate of the min-max values to obtain uniform energy upper bounds for almost every H > 0.
• Establish a Morse index estimate to control the index of the limit surface and obtain additional energy bounds in the presence of positive Ricci curvature.
• Use the CMC equation (1.1) and weak conformality condition (1.2) to characterize solutions as branched immersions with constant mean curvature H.
• Apply Sobolev inequalities and Poincaré-type estimates to control higher-order derivatives and ensure regularity in the min-max construction.

Experimental results

Research questions

• RQ1 Does there exist a branched immersed 2-sphere with constant mean curvature H > 0 in an arbitrary Riemannian 3-sphere?
• RQ2 Can the existence of such CMC 2-spheres be guaranteed for all H > 0 when the ambient 3-sphere has positive Ricci curvature?
• RQ3 Can a min-max construction for CMC surfaces be developed that ensures compactness and uniform energy bounds without relying on curvature pinching?
• RQ4 To what extent can the Sacks-Uhlenbeck min-max theory for minimal surfaces be generalized to the CMC setting in 3-spheres?

Key findings

• For almost every H > 0, there exists a nontrivial branched immersed 2-sphere with constant mean curvature H and Morse index at most 1 in any Riemannian 3-sphere diffeomorphic to S³.
• When the Riemannian 3-sphere has positive Ricci curvature, the existence of such a branched immersed CMC 2-sphere is guaranteed for every H > 0.
• The existence result holds under the weaker curvature condition Ric_g > -H²/2 g, extending beyond strict positive Ricci curvature.
• The Morse index of the constructed CMC 2-sphere is exactly 1 in the positive Ricci curvature case.
• The min-max construction achieves uniform energy bounds via derivative estimates and Morse index control, ensuring convergence to a nontrivial solution.
• The method generalizes the Sacks-Uhlenbeck existence theory for minimal 2-spheres to the CMC setting in 3-spheres.