Date of Award
Doctor of Philosophy
Alex Freire, Conrad Plaut, George Siopsis
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, . . . , κn) = 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.
Szapiel, Marek, "Hypersurfaces of prescribed curvature in hyperbolic space. " PhD diss., University of Tennessee, 2005.