Published Paper

Inserted: 7 oct 2022
Last Updated: 19 aug 2024

Journal: Discrete Contin. Dyn. Syst.
Year: 2023

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–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 \( n \geq 3 \). Using energy estimates for the linearized equation and a version of the Nash–Moser iteration, we demonstrate the local solvability for the fully nonlinear equation. Finally, we discuss some obstructions and perspectives on the global problem.

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• On-the-prescribed-negative-Gauss-curvature-problem-for-graphs.pdf