On the prescribed negative Gauss curvature problem for graphs

Abstract

<p style='text-indent:20px;'>We revisit the problem of prescribing negative Gauss curvature for graphs embedded in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{n+1} $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M2">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>. The problem reduces to solving a fully nonlinear Monge–Ampère equation that becomes hyperbolic in the case of negative curvature. We show that the linearization around a graph with Lorentzian Hessian can be written as a geometric wave equation for a suitable Lorentzian metric in dimensions <inline-formula><tex-math id="M3">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>. 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.</p>