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 Bj\"{o}rling formula for mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space Bull. Korean Math. Soc. 2017 Vol. 54, No. 1, 159-175 https://doi.org/10.4134/BKMS.b150984Published online January 31, 2017 Seong-Deog Yang Korea University Abstract : We solve the Bj\"{o}rling problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve $\gamma$ and an analytic (timelike in the case of de Sitter three-space) unit vector field $N$ along and orthogonal to $\gamma$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature $1$ which contains $\gamma$ and the unit normal of which on $\gamma$ is $N$. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC $1$ surfaces. Keywords : Bj\"{o}rling formula, constant mean curvature surfaces, de Sitter space MSC numbers : 53B30 Full-Text :