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Stability of Diophantine Minimal Foliations in Arbitrary Codimension
Abstract
We generalize Moser's stability theorem for minimal hypersurface foliations to arbitrary codimension. Let 𝕋N=ℝN/ℤN be the standard torus, and let P⊂ℝN be an m-plane satisfying a Diophantine condition relative to ℤN. We prove that, under every sufficiently small smooth ℤN-periodic perturbation of the Euclidean metric, the affine foliation of 𝕋N induced by P persists as a smoothly conjugate foliation by minimal m-dimensional submanifolds. The normalized conjugating displacement depends smoothly and tamely on the metric and is locally unique. The proof uses Hamilton's formulation of the Nash-Moser implicit function theorem.
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BibTeX
@misc{nguyen2026stability,
title = {{Stability of Diophantine Minimal Foliations in Arbitrary Codimension}},
author = {Hoan Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15984},
eprint = {2610.15984},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15984},
}
RIS
TY - UNPB
TI - Stability of Diophantine Minimal Foliations in Arbitrary Codimension
AU - Hoan Nguyen
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15984
UR - https://arxiv.org/abs/2610.15984
AB - We generalize Moser's stability theorem for minimal hypersurface foliations to arbitrary codimension. Let 𝕋^N=ℝ^N/ℤ^N be the standard torus, and let P⊂ℝ^N be an m-plane satisfying a Diophantine condition relative to ℤ^N. We prove that, under every sufficiently small smooth ℤ^N-periodic perturbation of the Euclidean metric, the affine foliation of 𝕋^N induced by P persists as a smoothly conjugate foliation by minimal m-dimensional submanifolds. The normalized conjugating displacement depends smoothly and tamely on the metric and is locally unique. The proof uses Hamilton's formulation of the Nash-Moser implicit function theorem.
ER -