论文标题
大型共形指标，带有处方的高斯和地球曲率
Large conformal metrics with prescribed Gaussian and geodesic curvatures
论文作者
论文摘要
我们考虑了为单位磁盘上的共形度量规定的高斯和测地曲线的问题。这等同于解决以下P.D.E. \begin{equation*}\begin{cases}-Δu=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_νu+2=2h(z)e^\frac u2＆\ hbox {on} \; \ partial \ mathbb {d}^2，\ end {cases} \ end {equation*}其中$ k，h $是处方的曲率。我们构建了一个带有曲率$ k_ \ varepsilon的共形指标家族，h_ \ varepsilon $分别收敛到$ k，h $，为$ \ varepsilon $ to $ 0 $，在某些通用假设下在一个边界点上升起。
We consider the problem of prescribing Gaussian and geodesic curvatures for a conformal metric on the unit disk. This is equivalent to solving the following P.D.E. \begin{equation*}\begin{cases}-Δu=2K(z)e^u&\hbox{in}\;\mathbb{D}^2,\\ \partial_νu+2=2h(z)e^\frac u2&\hbox{on}\;\partial\mathbb{D}^2,\end{cases} \end{equation*} where $K,h$ are the prescribed curvatures. We construct a family of conformal metrics with curvatures $K_\varepsilon,h_\varepsilon$ converging to $K,h$ respectively as $\varepsilon$ goes to $0$, which blows up at one boundary point under some generic assumptions.
