Hierarchy structures in finite index CMC surfaces

Abstract Given ε 0 > 0 {{\varepsilon}_{0}>0} , I ∈ ℕ ∪ { 0 } {I\in\mathbb{N}\cup\{0\}} and K 0 , H 0 ≥ 0 {K_{0},H_{0}\geq 0} , let X be a complete Riemannian 3-manifold with injectivity radius Inj ⁡ ( X ) ≥ ε 0 {\operatorname{Inj}(X)\geq{\varepsilon}_{0}} and with the supremum of absolute sectional curvature at most K 0 {K_{0}} , and let M ↬ X {M\looparrowright X} be a complete immersed surface of constant mean curvature H ∈ [ 0 , H 0 ] {H\in[0,H_{0}]} with index at most I. For such M ↬ X {M\looparrowright X} , we prove a structure theorem 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.