Hypersurfaces of prescribed curvature in hyperbolic space

Author

Marek Szapiel, University of Tennessee, Knoxville

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 ∈ 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.

Recommended Citation

Szapiel, Marek, "Hypersurfaces of prescribed curvature in hyperbolic space. " PhD diss., University of Tennessee, 2005.
https://trace.tennessee.edu/utk_graddiss/4185