On the prescribed negative Gauss curvature problem for graphs

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