论文标题
$ \ mathbb {h}^{n+1} $中规定的高斯曲率问题的流动方法
A flow approach to the prescribed Gaussian curvature problem in $\mathbb{H}^{n+1}$
论文作者
论文摘要
在本文中，我们研究以下规定的高斯曲率问题$$ k = \ frac {\ tilde {f}（θ）} {ϕ（ρ） （$α= n+1 $）在双曲线空间中，其中$ \ tilde {f} $是$ \ mathbb {s}^{n} $，$ρ$的平滑正函数，是高超曲面的径向函数，$ ϕ（ρ）= \sinhρ$ and $ k $是Gauss Curvature。通过流动方法，当$α\ geq n+1 $ $时，我们获得了上述方程的解决方案的存在和唯一性。我们的参数为$ \ Mathbb {H}^{n+1} $中的Alexandrov问题提供了平滑类别的抛物线证明。我们还考虑$ 2 <α\ leq n+1 $在$ \ tilde {f} $的均匀度下，并证明存在对上述方程式的解决方案。
In this paper, we study the following prescribed Gaussian curvature problem $$K=\frac{\tilde{f}(θ)}{ϕ(ρ)^{α-2}\sqrt{ϕ(ρ)^2+|\bar{\nabla}ρ|^2}},$$ a generalization of the Alexandrov problem ($α=n+1$) in hyperbolic space, where $\tilde{f}$ is a smooth positive function on $\mathbb{S}^{n}$, $ρ$ is the radial function of the hypersurface, $ϕ(ρ)=\sinhρ$ and $K$ is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when $α\geq n+1$. Our argument provides a parabolic proof in smooth category for the Alexandrov problem in $\mathbb{H}^{n+1}$. We also consider the cases $2<α\leq n+1$ under the evenness assumption of $\tilde{f}$ and prove the existence of solutions to the above equations.
