  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.
