  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.
