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

TL;DR

This paper establishes the existence of a smooth, complete 3-convex hypersurface in hyperbolic 5-space satisfying a specific prescribed curvature equation with given asymptotic boundary conditions, using a novel Lagrange multiplier approach.

Contribution

It introduces a Lagrange multiplier method to analyze the extremal concavity of a curvature function during global curvature estimates for hypersurfaces.

Findings

Existence of a smooth 3-convex hypersurface with prescribed curvature in hyperbolic space.

Development of a new Lagrange multiplier technique for curvature estimates.

Application to asymptotic Plateau problem in hyperbolic geometry.

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.