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We show that the Hopf differentials of a pair of isometric cousin surfaces, a minimal surface in euclidean 3-space and a constant mean curvature (CMC) one surface in the 3-dimensional hyperbolic space, with properly embedded annular ends, extend holomorphically to each end. Using this result, we derive conditions for when the pair must be a plane and a horosphere. Mostramos que as diferenciais de Hopf de um par de superfícies primas, a saber, uma superfície mínima em um espaço euclideano de dimensão 3 e uma superfície de curvatura média constante (CMC) um em um espaço hiperbólico de dimensão 3, se estendem holomorficamente em cada fim. Usando este resultado, obtemos condições para que o par seja um plano e uma horosfera

Minimal surfaces in euclidean 3-space and their mean curvature 1 cousins in hyperbolic 3-space