Hierarchy structures in finite index CMC surfaces

2022/12/27 by William H. Meeks, Meeks, William H., Joaquín Pérez +1
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2212.13594

Abstract

Given ε₀>0, I∈ ℕ∪ \0\ and K₀,H₀≥0, let X be a complete Riemannian 3-manifold with injectivity radius Inj(X)≥ ε₀ and with the supremum of absolute sectional curvature at most K₀, and let M \looparrowright X be a complete immersed surface of constant mean curvature H∈ [0,H₀] with index at most I. For such M \looparrowright X, we prove Structure Theorem 1.2 which describes how the interesting ambient geometry of the immersion is organized locally around at most I points of M where the norm of the second fundamental form takes on large local maximum values.

Related

• Geometry of CMC surfaces of finite index
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• On the Index of Constant Mean Curvature Hypersurfaces
• (CMC) 1-immersions of surfaces into hyperbolic 3-manifolds
• Radius estimates for nearly stable H-hypersurfaces of dimension 2, 3, and 4