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Meeks III","submitted_at":"2016-03-07T15:39:51Z","abstract_excerpt":"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 $2\\pi \\chi(\\Sigma)$. 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