Hypersurfaces of prescribed curvature in hyperbolic space

In this paper we consider the problem of existence of hypersurfaces with prescribed curvature in hyperbolic space. We use the upper half-space model of hyperbolic space. The hypersurfaces we consider are given as graphs of positive functions on some domain Ω ∈ Rⁿ satisfying equations of form

f (A) = f (κ₁, . . . , κ_n) = ψ,

where A is the second fundamental form of a hypersurface, f (A) is a smooth sym- metric function of the eigenvalues of A and ψ is a function of position. If we impose certain conditions on f and ψ, the above equation can be treated as an elliptic, fully non-linear partial differential equation

G(D²u, Du, u) = ψ(x, u).

We then derive an existence result for the corresponding Dirichlet problem.