  The problem of the prescribed curvature measure is one of the important
problems in differential geometry and nonlinear partial differential equations.
In this paper, we consider prescribed curvature measure problem in hyperbolic
space. We obtain the existence of star-shaped k-convex bodies with prescribed
(n-k)-th curvature measures (k<n) by establishing crucial C^2 regularity
estimates for solutions to the corresponding fully nonlinear PDE in the
hyperbolic space.
