Year: 2026
Type: preprint
Abstract: We solve the asymptotic Plateau problem in hyperbolic space for the curvature given by the normalized geometric mean of the $p$-fold sums of the principal curvatures. For every $n\ge3$, $2\le p\le n-1$, and $σ\in(0,1)$, each bounded domain $Ω\subset\mathbb{R}^n$ with smooth mean-convex boundary admits a unique complete admissible vertical graph with prescribed curvature $σ$ and asymptotic boundary $\partialΩ\times\{0\}$. The result extends the known cases $(n,p)=(3,2)$ and $(4,3)$ to the stated range. The key curvature estimate is uniform with respect to the positive boundary height in the approximating Dirichlet problems. The proof establishes two lower bounds for the full third-derivative quadratic form under the linearized constraints: a sharp universal bound with coefficient $2/p$ and a stronger direction-dependent bound when the relevant principal curvature lies in a specified range.
Source: arXiv (Cornell University)
Authors: Shi, Shujun, Sui, Zhenan
Institution: Harbin Institute of Technology
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Award: G5665659986
OpenAlex ID: W7214795462