On the prescribed negative Gauss curvature problem for graphs

Alessio Figalli; Christoph Kehle

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arXiv · 2209.02326

On the prescribed negative Gauss curvature problem for graphs

Alessio Figalli·Christoph Kehle

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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-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.

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Alessio Figalli, Christoph Kehle. 2022-09-06. On the prescribed negative Gauss curvature problem for graphs. https://arxiv.org/abs/2209.02326

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