论文标题
关于图形的规定的负高斯曲率问题
On the prescribed negative Gauss curvature problem for graphs
论文作者
论文摘要
我们重新审查嵌入在$ \ Mathbb r^{n+1} $中的图形的负高曲率的问题时，当$ n \ geq 2 $。该问题减少了解决完全非线性的monge-ampère方程，该方程在负曲率的情况下变成双曲线。我们表明，与Lorentzian Hessian围绕图的线性化可以写为适合Lorentzian尺寸的几何波动方程，$ n \ geq 3 $。使用线性化方程式和NASH-MOSER迭代版本的能量估计，我们显示了完全非线性方程的局部溶解性。最后，我们讨论了有关全球问题的一些障碍和观点。
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 show the local solvability for the fully nonlinear equation. Finally, we discuss some obstructions and perspectives on the global problem.
