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

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.

minimal surfaces; CMC 1 cousins; hyperbolic space


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