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Area gaps for minimal surfaces in spheres
Abstract
In this paper, we prove that a linearly full minimal $2$-torus in πN with Nβ₯ 5 must have area >2Ο2, and that a linearly full closed minimal surface in πN with Nβ₯ 9 must have area >8Ο. These results verify the Willmore conjecture and the generalized Willmore conjecture in higher codimension for minimal surfaces. The area gaps are obtained by applying the Cheng--Li--Yau comparison theorem and deriving a lower bound of the heat trace, where the monotonicity of the shifted normalized Riesz mean is used. We also establish a volume gap above the balanced Clifford product for linearly full closed minimal n-submanifolds in πN when n=3, Nβ₯ 9, or nβ₯ 4, Nβ₯ 3nβ1.
Related
- High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres
- Area-minimizing submanifolds are not generically smooth, except for geodesics, minimal surfaces, and minimal hypersurfaces
- Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds
- Mass Bounds for Confined Area-Minimizing Minimal Surfaces
- Rigidity of Minimal Submanifolds in Spheres of Higher Codimension
BibTeX
@misc{wang2026area,
title = {{Area gaps for minimal surfaces in spheres}},
author = {Peng Wang and Zhenxiao Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28317},
eprint = {2610.28317},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28317},
}
RIS
TY - UNPB
TI - Area gaps for minimal surfaces in spheres
AU - Peng Wang
AU - Zhenxiao Xie
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28317
UR - https://arxiv.org/abs/2610.28317
AB - In this paper, we prove that a linearly full minimal $2$-torus in π^N with Nβ₯ 5 must have area >2Ο^2, and that a linearly full closed minimal surface in π^N with Nβ₯ 9 must have area >8Ο. These results verify the Willmore conjecture and the generalized Willmore conjecture in higher codimension for minimal surfaces. The area gaps are obtained by applying the Cheng--Li--Yau comparison theorem and deriving a lower bound of the heat trace, where the monotonicity of the shifted normalized Riesz mean is used. We also establish a volume gap above the balanced Clifford product for linearly full closed minimal n-submanifolds in π^N when n=3, Nβ₯ 9, or nβ₯ 4, Nβ₯ 3nβ1.
ER -