On the prescribed negative Gauss curvature problem for graphs

TL;DR

This paper investigates the problem of prescribing negative Gauss curvature for graphs in higher dimensions, transforming it into a hyperbolic Monge-Ampère equation and establishing local solvability through geometric wave equations and Nash-Moser iteration.

Contribution

It introduces a novel approach by linearizing around Lorentzian Hessians, linking the problem to geometric wave equations, and proving local solutions for the nonlinear PDE.

Findings

Linearization yields a geometric wave equation in Lorentzian metrics.

Local solvability established via energy estimates and Nash-Moser iteration.

Discussion of obstructions and future perspectives on the global problem.

Abstract

We revisit the problem of prescribing negative Gauss curvature for graphs embedded in $\mathbb R^{n+1}$ when $n\geq 2$. The problem reduces to solving a fully nonlinear Monge-Amp\`ere 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 $n\geq 3$. 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.