#### Date of Award

8-2005

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy

#### Major

Mathematics

#### Major Professor

Bo Guan

#### Committee Members

Alex Freire, Conrad Plaut, George Siopsis

#### Abstract

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 ∈ **R**^{n} satisfying equations of form

*f (A) = f (κ _{1}, . . . , 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(D ^{2}u, Du, u) = ψ(x, u).*

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

#### Recommended Citation

Szapiel, Marek, "Hypersurfaces of prescribed curvature in hyperbolic space. " PhD diss., University of Tennessee, 2005.

http://trace.tennessee.edu/utk_graddiss/4185