标题： 渐近Plateau问题对于$2$-凸超曲面在$\mathbb{H}^4$中的情况 显示英文标题

标题： Asymptotic Plateau problem for $2$-convex hypersurface in $\mathbb{H}^4$

摘要： 我们证明了存在光滑完备的$2$-凸超曲面，它满足预设的曲率方程$(\kappa_1 + \kappa_2)(\kappa_1 + \kappa_3)(\kappa_2 + \kappa_3) = (2 \sigma)^3$，并且在四维双曲空间无穷远处具有预设的渐近边界，其中$\sigma \in (0, 1)$是常数。 我们还证明了在$\mathbb{H}^{n + 1}$中$\sigma_k (\kappa_2 + \cdots + \kappa_n, \cdots, \kappa_1 + \cdots + \kappa_{n - 1}) = C_n^k (n - 1)^k \sigma^k$关于$k < n$的存在性。 显示英文摘要

摘要： We prove the existence of smooth complete $2$-convex hypersurface which satisfies prescribed curvature equation $(\kappa_1 + \kappa_2)(\kappa_1 + \kappa_3)(\kappa_2 + \kappa_3) = (2 \sigma)^3$ and has prescribed asymptotic boundary at infinity of hyperbolic space of dimension 4, where $\sigma \in (0, 1)$ is a constant. We also prove the existence for $\sigma_k (\kappa_2 + \cdots + \kappa_n, \cdots, \kappa_1 + \cdots + \kappa_{n - 1}) = C_n^k (n - 1)^k \sigma^k$ with $k < n$ in $\mathbb{H}^{n + 1}$.