BJORLING FORMULA FOR MEAN CURVATURE ONE SURFACES IN HYPERBOLIC THREE-SPACE AND IN DE SITTER THREE-SPACE

We solve the Bjorling 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.