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PreprintCC BY 4.0arXiv

Mass Bounds for Confined Area-Minimizing Minimal Surfaces

  • Xumin Jiang
  • Jiongduo Xie
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Abstract

We establish codimension-independent mass bounds for geometrically confined area-minimizing rectifiable currents. In Euclidean space, we combine the confined-volume doubling theorem of Colding--Minicozzi with a current-theoretic squashing argument. This gives an affirmative answer to Lin's interior mass-bound problem for every algebraic projection multiplicity Q: the interior mass is bounded by C(n)Q, without an a priori mass bound at a larger scale. This result yields a degree-one Bernstein-type rigidity result under sublinear confinement. For the hyperbolic application, we make the curvature modification of the fixed-scale argument needed in a thin tubular neighborhood of a totally geodesic copy of ℍn. Combining this auxiliary estimate with a localized squashing estimate removes the doubly exponential local mass-growth condition from the boundary regularity results in [13].

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Cite

BibTeXRIS
BibTeX
@misc{jiang2026mass,
  title = {{Mass Bounds for Confined Area-Minimizing Minimal Surfaces}},
  author = {Xumin Jiang and Jiongduo Xie},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.01324},
  eprint = {2609.01324},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.01324},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Mass Bounds for Confined Area-Minimizing Minimal Surfaces
AU  - Xumin Jiang
AU  - Jiongduo Xie
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.01324
UR  - https://arxiv.org/abs/2609.01324
AB  - We establish codimension-independent mass bounds for geometrically confined area-minimizing rectifiable currents. In Euclidean space, we combine the confined-volume doubling theorem of Colding--Minicozzi with a current-theoretic squashing argument. This gives an affirmative answer to Lin's interior mass-bound problem for every algebraic projection multiplicity Q: the interior mass is bounded by C(n)Q, without an a priori mass bound at a larger scale. This result yields a degree-one Bernstein-type rigidity result under sublinear confinement. For the hyperbolic application, we make the curvature modification of the fixed-scale argument needed in a thin tubular neighborhood of a totally geodesic copy of ℍ^n. Combining this auxiliary estimate with a localized squashing estimate removes the doubly exponential local mass-growth condition from the boundary regularity results in [13].
ER  -