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A min-max gap characterization of minimal foliations on the torus

  • Hoan Nguyen
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Abstract

We extend an energy introduced by Mather to the setting of Almgren-Pitts min-max theory and obtain a parametric, higher-dimensional analogue of Mather's variational barrier theory for twist maps and geodesics on tori. We use this energy to establish several criteria for the existence of foliations of the n-torus by minimal hypersurfaces. We show that for a generic metric, whenever a lamination by area-minimizing hypersurfaces of the n-torus contains a gap, there exists a minimal hypersurface inside the gap that is not area-minimizing. This hypersurface is a higher-dimensional analogue of the secondary minimax orbit appearing in Aubry-Mather theory.

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Cite

BibTeXRIS
BibTeX
@misc{nguyen2026minmax,
  title = {{A min-max gap characterization of minimal foliations on the torus}},
  author = {Hoan Nguyen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.12428},
  eprint = {2605.12428},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.12428},
}
RIS
TY  - UNPB
TI  - A min-max gap characterization of minimal foliations on the torus
AU  - Hoan Nguyen
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2605.12428
UR  - https://arxiv.org/abs/2605.12428
AB  - We extend an energy introduced by Mather to the setting of Almgren-Pitts min-max theory and obtain a parametric, higher-dimensional analogue of Mather's variational barrier theory for twist maps and geodesics on tori. We use this energy to establish several criteria for the existence of foliations of the n-torus by minimal hypersurfaces. We show that for a generic metric, whenever a lamination by area-minimizing hypersurfaces of the n-torus contains a gap, there exists a minimal hypersurface inside the gap that is not area-minimizing. This hypersurface is a higher-dimensional analogue of the secondary minimax orbit appearing in Aubry-Mather theory.
ER  -