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A Frankel type theorem in Euclidean and hyperbolic spaces
Abstract
We prove that a connected mean convex region in ℝn+1 with at least two boundary components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature must decay at infinity, and generalize this result to hyperbolic space.
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BibTeX
@misc{xue2026frankel,
title = {{A Frankel type theorem in Euclidean and hyperbolic spaces}},
author = {Haotian Xue},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.19679},
eprint = {2605.19679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.19679},
}
RIS
TY - UNPB
TI - A Frankel type theorem in Euclidean and hyperbolic spaces
AU - Haotian Xue
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2605.19679
UR - https://arxiv.org/abs/2605.19679
AB - We prove that a connected mean convex region in ℝ^n+1 with at least two boundary components cannot have strictly positive mean curvature. This answers a question of Gromov. We also obtain estimates for how quickly the mean curvature must decay at infinity, and generalize this result to hyperbolic space.
ER -