On the prescribed negative Gauss curvature problem for graphs

Abstract

We revisit the problem of prescribing negative Gauss curvature for graphs embedded in ${\mathbb{R}}^{n+1}$ when $n\ge 2$. The problem reduces to solving a fully nonlinear Monge-Ampère equation that becomes hyperbolic in the case of negative curvature. We show that the linearization around a graph with Lorentzian Hessian can be written as a geometric wave equation for a suitable Lorentzian metric in dimensions $n\ge 3$. Using energy estimates for the linearized equation and a version of the Nash-Moser iteration, we show the local solvability for the fully nonlinear equation. Finally, we discuss some obstructions and perspectives on the global problem.

arXiv comment17 pages, 2 figures, to appear in Discrete Contin. Dyn. Syst