Björling formula for mean curvature one surfaces in hyperbolic three-space and in de sitter three-space

We solve the Björling 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 and an analytic (timelike in the case of de Sitter three-space) unit vector field N along and orthogonal to γ, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains γ and the unit normal of which on γ is N. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.