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 $\prod\limits_{i = 1}^n (H - \kappa_i) = \big( (n - 1) \sigma \big)^n$ for $n = 4$ and has prescribed asymptotic boundary $\Gamma$ at the infinity of hyperbolic space of dimension 5, where $\sigma \in (0, 1)$ is a constant and $\Gamma$ is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of $f (\kappa) = \frac{1}{n - 1} \Big( \prod\limits_{i = 1}^n (H - \kappa_i) \Big)^{\frac{1}{n}}$ during uniform global curvature estimate.