Geometry of CMC surfaces of finite index

TL;DR

This paper investigates the geometric properties of constant mean curvature surfaces with finite index in 3-manifolds, providing bounds on their area and diameter, and establishing conditions for their compactness and genus growth.

Contribution

It introduces new area and diameter estimates for CMC surfaces of finite index, extending previous results and applying the Hierarchy Structure Theorem for global geometric analysis.

Findings

Area grows linearly with genus for connected surfaces.

Large genus implies a lower bound on area depending on curvature bounds.

Under positive scalar curvature conditions, surfaces are compact with bounded area and diameter.

Abstract

Given $r_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\geq 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $\mbox{Inj}(X)\geq r_0$ 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 $H\in [0,H_0]$ and with index at most $I$. We will obtain geometric estimates for such an $M\looparrowright X$ as a consequence of the Hierarchy Structure Theorem in [9]. The Hierarchy Structure Theorem (see Theorem 2.2 below) will be applied to understand global properties of $M\looparrowright X$, especially results related to the area and diameter of $M$. By item E of Theorem 2.2, the area of such a non-compact $M\looparrowright X$ is infinite. We will improve this area result by proving the following when $M$ is connected; here $g(M)$ denotes the genus of the orientable cover of $M$: 1. There exists $C_1=C_1(I,r_0,K_0,H_0)>0$ such that Area$(M)\geq C_1(g(M)+1)$. 2. There exists $C>0,G(I)\in \mathbb{N}$ independent of $r_0,K_0,H_0$ and also $C$ independent of $I$ such that if $g(M)\geq G(I)$, then Area$(M)\geq \frac{C}{(\max\{1,\frac{1}{r_0},\sqrt{K_0}, H_0\})^2}(g(M)+1)$. 3. If the scalar curvature $\rho$ of $X$ satisfies $3H^2+\frac{1}{2}\rho\geq c$ in $X$ for some $c>0$, then there exist $A,D>0$ depending on $c,I,r_0,K_0,H_0$ such that Area$(M)\leq A$ and Diameter$(M)\leq D$. Hence, $M$ is compact and, by item 1, $g(M)\leq A/C -1$.