Hypersurfaces of constant scalar curvature in hyperbolic space with prescribed asymptotic boundary at infinity

In this note, we study the asymptotic Plateau problem in hyperbolic space, and we prove the existence of a smooth complete hypersurface of constant scalar curvature in hyperbolic space with a prescribed asymptotic boundary at infinity. Following a pioneering work of Bo Guan and Joel Spruck, we seek the solution as a vertical graph over a bounded domain and solve the corresponding Dirichlet problem for a fully nonlinear partial differential equation by establishing the crucial second order estimates for admissible solutions. Our proof consists of three main ingredients: (1) a non-standard, special choice for the parameter in the auxiliary function, (2) a so-called almost-Jacobi inequality for the equation operator, and (3) a set of arguments which reduce the situation to semi-convex case and which keep the coefficient of the troublesome negative term within a suitable magnitude.