A dichotomy for minimal submanifolds
Abstract
We survey a circle of recent results showing that a minimal submanifold obeys a dichotomy: either it fills up space, spreading out like a space-filling curve, or it is confined, and confinement forces quantitative restrictions. On the confined side these restrictions are both geometric and function-theoretic: Euclidean volume growth, an optimal rate of convergence of the density, complex-curve rigidity for stable surfaces in bfR4, and a Liouville theorem forcing slowly growing harmonic functions on minimal disks to be constant. A prototypical restriction is Euclidean volume growth, which in these results is forced by the geometry rather than assumed. The mechanism is a volume doubling theorem that converts geometric confinement into quantitative rigidity: a stationary integral varifold trapped in a thin slab at a given scale cannot double its volume by more than a universal factor. We explain how this principle, and the height-excess bounds behind it, produce Euclidean volume growth for submanifolds of sublinearly growing height in every dimension and codimension, the optimal density rate in a slab, the complex-curve and Liouville rigidity above, a higher-codimension Bernstein theorem for disks, one-sided volume bounds, and an optimal stable Bernstein theorem in all dimensions generalizing Moser, Bombieri-De Giorgi-Miranda, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. We describe three regimes with sharp borderlines. In narrow cones the volume grows polynomially with exponent n+O(ε2), and the order ε2 is attained; in wide cones the growth can be arbitrarily fast. For holomorphic curves in ℂ2, there are also three regimes and the borderline is determined by the Kähler angle.
Related
- Rigidity of Minimal Submanifolds in Spheres of Higher Codimension
- Orientable minimal submanifolds of low index in the sphere
- The pinching constant for closed minimal submanifolds of high codimension in the sphere
- A Curvature Gap for Minimal Submanifolds in Spheres
- Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
BibTeX
@misc{colding2026dichotomy,
title = {{A dichotomy for minimal submanifolds}},
author = {Tobias Holck Colding and William P. Minicozzi II},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.29502},
eprint = {2607.29502},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.29502},
}
RIS
TY - UNPB
TI - A dichotomy for minimal submanifolds
AU - Tobias Holck Colding
AU - William P. Minicozzi II
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2607.29502
UR - https://arxiv.org/abs/2607.29502
AB - We survey a circle of recent results showing that a minimal submanifold obeys a dichotomy: either it fills up space, spreading out like a space-filling curve, or it is confined, and confinement forces quantitative restrictions. On the confined side these restrictions are both geometric and function-theoretic: Euclidean volume growth, an optimal rate of convergence of the density, complex-curve rigidity for stable surfaces in bfR^4, and a Liouville theorem forcing slowly growing harmonic functions on minimal disks to be constant. A prototypical restriction is Euclidean volume growth, which in these results is forced by the geometry rather than assumed. The mechanism is a volume doubling theorem that converts geometric confinement into quantitative rigidity: a stationary integral varifold trapped in a thin slab at a given scale cannot double its volume by more than a universal factor. We explain how this principle, and the height-excess bounds behind it, produce Euclidean volume growth for submanifolds of sublinearly growing height in every dimension and codimension, the optimal density rate in a slab, the complex-curve and Liouville rigidity above, a higher-codimension Bernstein theorem for disks, one-sided volume bounds, and an optimal stable Bernstein theorem in all dimensions generalizing Moser, Bombieri-De Giorgi-Miranda, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. We describe three regimes with sharp borderlines. In narrow cones the volume grows polynomially with exponent n+O(ε^2), and the order ε^2 is attained; in wide cones the growth can be arbitrarily fast. For holomorphic curves in ℂ^2, there are also three regimes and the borderline is determined by the Kähler angle.
ER -