We study complete finite topology immersed surfaces $\Sigma$ in complete Riemannian $3$-manifolds $N$ with sectional curvature $K_N\leq -a^2\leq 0$, such that the absolute mean curvature function of $\Sigma$ is bounded from above by $a$ and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface $\Sigma$ must be proper in $N$ and its total curvature must be equal to $2\pi \chi(\Sigma)$. If $N$ is a hyperbolic $3$-manifold of finite volume and $\Sigma$ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of $\Sigma$ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in $N$.

中文翻译：

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正确浸入双曲线 $3$-歧管中的表面

我们在具有截面曲率 $K_N\leq -a^2\leq 0$ 的完全黎曼 $3$-流形 $N$ 中研究完全有限拓扑浸没面 $\Sigma$，使得 $\Sigma$ 的绝对平均曲率函数为从上方以$a$为界，并且其注入半径函数在其每个环形末端代表上均不以零为界。我们证明这样的曲面$\Sigma$ 必须在$N$ 中是适当的，并且它的总曲率必须等于$2\pi\chi(\Sigma)$。如果 $N$ 是有限体积的双曲 $3$-流形，$\Sigma$ 是一个非负常数平均曲率小于 1 的有限拓扑的适当浸入表面，那么我们证明 $\Sigma$ 的每一端是渐近的（具有有限正多重性）到完全脐带环，正确嵌入 $N$。