The Cauchy problem for Liouville equation and Bryant surfaces

We give a construction that connects the Cauchy problem for Liouville elliptic equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H3, and solve both of them. We construct the only mean curvature one surface in H3 that passes through a given curve with given unit normal along it, and provide diverse applications. In particular, topics like period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. of surfaces are treated.