Asymptotic Plateau problem for $3$-convex hypersurface in ${\mathbb{H}}^5$

Abstract

We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod _{i=1}^n(H-\hat{I}{º}_i)=((n-1)\ddot{I}ƒ{)}^n$ for $n=4$ and has prescribed asymptotic boundary $\hat{I}“$ at the infinity of hyperbolic space of dimension 5, where $\ddot{I}ƒ\in (0,1)$ is a constant and $\hat{I}“$ is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of $f(\hat{I}º)=\frac{1}{n-1}(\prod _{i=1}^n(H-\hat{I}{º}_i){)}^{\frac{1}{n}}$ during uniform global curvature estimate.

This version is a follow up work of the previous version. The previous version is for dimension $n=3$, and this version is for $n=4$. They are indeed two different papers