Calculus of Variations and Geometric Measure Theory

A. Figalli - C. Kehle

On the prescribed negative Gauss curvature problem for graphs

created by figalli on 07 Oct 2022


Accepted Paper

Inserted: 7 oct 2022
Last Updated: 7 oct 2022

Journal: Discrete Contin. Dyn. Syst.
Year: 2022


We revisit the problem of prescribing negative Gauss curvature for graphs embedded in Rn+1 when n ≥ 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 ≥ 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.