The Asymptotic Plateau Problem for p-Convex Hypersurfaces in Hyperbolic Space
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≥3, 2≤ p≤ n−1, and σ∈(0,1), each bounded domain Ω⊂ℝn with smooth mean-convex boundary admits a unique complete admissible vertical graph with prescribed curvature σ and asymptotic boundary ∂Ω×{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.
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BibTeX
@misc{shi2026asymptotic,
title = {{The Asymptotic Plateau Problem for $p$-Convex Hypersurfaces in Hyperbolic Space}},
author = {Shujun Shi and Zhenan Sui},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34751},
eprint = {2609.34751},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34751},
}
RIS
TY - UNPB
TI - The Asymptotic Plateau Problem for p-Convex Hypersurfaces in Hyperbolic Space
AU - Shujun Shi
AU - Zhenan Sui
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34751
UR - https://arxiv.org/abs/2609.34751
AB - 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≥3, 2≤ p≤ n−1, and σ∈(0,1), each bounded domain Ω⊂ℝ^n with smooth mean-convex boundary admits a unique complete admissible vertical graph with prescribed curvature σ and asymptotic boundary ∂Ω×{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.
ER -