Minimal foliations, codimension-one stable norms, and a question of Bangert
Abstract
We compute the codimension-one stable norm for a natural class of cohomogeneity-one metrics on tori. In every dimension n≥3, the formula yields smooth nonflat metrics for which each primitive codimension-one homology class is represented by a foliation of calibrated tori, giving a negative answer to a question of Bangert. On 𝕋3, we construct an infinite-dimensional family of nonflat metrics whose codimension-one stable norm agrees exactly with that of the unit cubic flat torus and whose total volume is fixed. An explicit two-parameter subfamily contains pairwise non-isometric metrics. These examples also show that the Euclidean-stable-norm-and-volume data are not locally injective near the cubic flat metric. Conversely, among smooth metrics on 𝕋3 admitting a free isometric circle action and having the cubic Euclidean codimension-one stable norm, we prove that volume is at most one, with equality only for the cubic flat metric up to an isometry isotopic to the identity.
Related
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- CMC foliations on Euclidean spaces are minimal foliations
BibTeX
@misc{nguyen2026minimal,
title = {{Minimal foliations, codimension-one stable norms, and a question of Bangert}},
author = {Hoan Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18428},
eprint = {2608.18428},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18428},
}
RIS
TY - UNPB
TI - Minimal foliations, codimension-one stable norms, and a question of Bangert
AU - Hoan Nguyen
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2608.18428
UR - https://arxiv.org/abs/2608.18428
AB - We compute the codimension-one stable norm for a natural class of cohomogeneity-one metrics on tori. In every dimension n≥3, the formula yields smooth nonflat metrics for which each primitive codimension-one homology class is represented by a foliation of calibrated tori, giving a negative answer to a question of Bangert. On 𝕋^3, we construct an infinite-dimensional family of nonflat metrics whose codimension-one stable norm agrees exactly with that of the unit cubic flat torus and whose total volume is fixed. An explicit two-parameter subfamily contains pairwise non-isometric metrics. These examples also show that the Euclidean-stable-norm-and-volume data are not locally injective near the cubic flat metric. Conversely, among smooth metrics on 𝕋^3 admitting a free isometric circle action and having the cubic Euclidean codimension-one stable norm, we prove that volume is at most one, with equality only for the cubic flat metric up to an isometry isotopic to the identity.
ER -