Minimal submanifolds confined in space
Abstract
Already in ℝ4, there are many minimal hypersurfaces but few structural results. The half-space theorem fails there: higher-dimensional catenoids are among the many examples contained in a slab. Nevertheless, we show that in any dimension and codimension, the height of a minimal submanifold controls its volume growth: confinement to a narrow cone forces polynomial growth, and sublinear height forces Euclidean growth, the slowest possible. Thus volume bounds, which regularity theory assumes, follow from height alone. The key is a volume doubling theorem that converts geometric confinement into quantitative rigidity: if an n-dimensional stationary integral varifold lies in a thin slab at a given scale, then its volume doubling ratio at that scale is bounded by a universal constant. Iterating this gives the polynomial growth bounds above. Sublinear height is exactly the borderline for Euclidean growth: examples grow at least like rn+4ε2/5 in cones of aperture ε. From Euclidean volume growth, we obtain an optimal Bernstein theorem, in all dimensions, for stable minimal hypersurfaces with sublinear height growth, generalizing results of Moser, Bombieri--De Giorgi--Miranda, Trudinger, Caffarelli--Nirenberg--Spruck, and Ecker--Huisken.
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{colding2026minimal,
title = {{Minimal submanifolds confined in space}},
author = {Tobias Holck Colding and William P. Minicozzi II},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.15031},
eprint = {2605.15031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.15031},
}
RIS
TY - UNPB
TI - Minimal submanifolds confined in space
AU - Tobias Holck Colding
AU - William P. Minicozzi II
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2605.15031
UR - https://arxiv.org/abs/2605.15031
AB - Already in ℝ^4, there are many minimal hypersurfaces but few structural results. The half-space theorem fails there: higher-dimensional catenoids are among the many examples contained in a slab. Nevertheless, we show that in any dimension and codimension, the height of a minimal submanifold controls its volume growth: confinement to a narrow cone forces polynomial growth, and sublinear height forces Euclidean growth, the slowest possible. Thus volume bounds, which regularity theory assumes, follow from height alone. The key is a volume doubling theorem that converts geometric confinement into quantitative rigidity: if an n-dimensional stationary integral varifold lies in a thin slab at a given scale, then its volume doubling ratio at that scale is bounded by a universal constant. Iterating this gives the polynomial growth bounds above. Sublinear height is exactly the borderline for Euclidean growth: examples grow at least like r^n+4ε^2/5 in cones of aperture ε. From Euclidean volume growth, we obtain an optimal Bernstein theorem, in all dimensions, for stable minimal hypersurfaces with sublinear height growth, generalizing results of Moser, Bombieri--De Giorgi--Miranda, Trudinger, Caffarelli--Nirenberg--Spruck, and Ecker--Huisken.
ER -