We prove the existence of a smooth complete hypersurfaces of constant scalar curvature in hyperbolic space, with a prescribed asymptotic boundary at infinity. Following a pioneering work of Guan-Spruck, we seek the solution as a graph over a bounded domain and solve the corresponding Dirichlet problem by establishing the crucial second order estimates for admissible solutions.
Our proof consists of three main ingredients: (1) a new test function, (2) the almost-Jacobi inequality due to Shankar-Yuan, and (3) a new set of arguments which reduce the situation to semi-convex case.
