On the prescribed negative Gauss curvature problem for graphs
| Autor: | Figalli, Alessio, Kehle, Christoph |
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| Jazyk: | angličtina |
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Discrete and Continuous Dynamical Systems, 43 (3-4) |
| Popis: | 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. Comment: 17 pages, 2 figures, to appear in Discrete Contin. Dyn. Syst |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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