Geometry of CMC surfaces of finite index

William H. Meeks III, Joaquin Perez

Published 2022-12-29Version 1

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 with index at most $I$ of constant mean curvature $H\in [0,H_0]$. We will obtain geometric estimates for such an $M\looparrowright X$ as a consequence of the Hierarchy Structure Theorem in [3]. 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 $C=C(I,r_0,K_0,H_0)>0$ such that Area$(M)\geq C(g(M)+1)$. 2. There exists $C'>0$ independent of $I,r_0,K_0,H_0$ and $G=G(I,r_0,K_0,H_0)\in \mathbb{N}$ such that if $g(M)\geq G$, then Area$(M)\geq C'(g(M)+1)$. 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$.