Hierarchy structures in finite index CMC surfaces

Given ${\epsilon }_0>0$, $I\in \mathbb{N}\cup \{0\}$ and $K_0,H_0\ge 0$, let $X$ be a complete Riemannian $3$-manifold with injectivity radius $Inj(X)\ge {\epsilon }_0$ 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.