Hierarchy structures in finite index CMC surfaces

TL;DR

This paper investigates the local geometric structure of finite index constant mean curvature surfaces within 3-manifolds, revealing how their complex ambient geometry is organized around finitely many points with large curvature.

Contribution

It establishes a structure theorem detailing the organization of ambient geometry near points of high curvature on finite index CMC surfaces in 3-manifolds.

Findings

Describes local organization of ambient geometry around high curvature points.

Provides bounds on the number of such points based on index.

Characterizes the structure of CMC surfaces in bounded curvature manifolds.

Abstract

Given $\varepsilon_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq \varepsilon_0$ and with the supremum of absolute sectional curvature at most $K_0$, and let $M \looparrowright X$ be a complete immersed surface of constant mean curvature $H\in [0,H_0]$ 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.