Minimal translation hypersurfaces in hyperbolic space
Abstract
In the upper half-space model of the hyperbolic space ℍn+1, a translation hypersurface of type I is a graph xn+1=f1(x1)+⋯+fn(xn) and a translation hypersurface of type II is a graph xn=f1(x1)+⋯+fn−1(xn−1)+g(xn+1). We classify the minimal translation hypersurfaces of both types for every n≥ 2. We obtain these results as particular cases of a classification of the singular minimal translation hypersurfaces of ℝn+1 for an arbitrary exponent α≠ 0, with respect to a direction orthogonal (type I) or parallel (type II) to the hyperplane over which the hypersurface is a graph. The minimal translation hypersurfaces of type I are exactly the parabolic cylinders over a one-parameter family of arches, which contradicts a nonexistence theorem in the literature. The minimal translation hypersurfaces of type II are the parabolic cylinders, the vertical totally geodesic hyperplanes and, when n≥ 3, the vertical cylinders over the Scherk surface.
Related
- Strongly stable CMC-one hypersurfaces in every hyperbolic space of dimension at least four
- Hypersurfaces of constant higher order mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity
- The asymptotic Plateau problem for Hypersurfaces of constant Hk curvature in hyperbolic space
- Free boundary minimal hypersurfaces of Morse index one in Eucildean half-space
- The Asymptotic Plateau Problem for p-Convex Hypersurfaces in Hyperbolic Space
BibTeX
@misc{ndiaye2026minimal,
title = {{Minimal translation hypersurfaces in hyperbolic space}},
author = {Ameth Ndiaye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10363},
eprint = {2610.10363},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10363},
}
RIS
TY - UNPB
TI - Minimal translation hypersurfaces in hyperbolic space
AU - Ameth Ndiaye
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10363
UR - https://arxiv.org/abs/2610.10363
AB - In the upper half-space model of the hyperbolic space ℍ^n+1, a translation hypersurface of type I is a graph x_n+1=f_1(x_1)+⋯+f_n(x_n) and a translation hypersurface of type II is a graph x_n=f_1(x_1)+⋯+f_n−1(x_n−1)+g(x_n+1). We classify the minimal translation hypersurfaces of both types for every n≥ 2. We obtain these results as particular cases of a classification of the singular minimal translation hypersurfaces of ℝ^n+1 for an arbitrary exponent α≠ 0, with respect to a direction orthogonal (type I) or parallel (type II) to the hyperplane over which the hypersurface is a graph. The minimal translation hypersurfaces of type I are exactly the parabolic cylinders over a one-parameter family of arches, which contradicts a nonexistence theorem in the literature. The minimal translation hypersurfaces of type II are the parabolic cylinders, the vertical totally geodesic hyperplanes and, when n≥ 3, the vertical cylinders over the Scherk surface.
ER -