PreprintCC BY 4.0arXiv
Biharmonic hypersurfaces in space forms
Abstract
We prove that every biharmonic hypersurface in a space form of nonpositive sectional curvature is minimal in arbitrary dimension. This settles the hypersurface case of the generalized Chen's conjecture in hyperbolic space and gives a unified treatment of nonpositive space forms. For biharmonic hypersurfaces in the unit sphere, we derive quantitative restrictions on every possible nonconstant-mean-curvature solution. In particular, we obtain pointwise criteria forcing constant mean curvature, a strict scalar-curvature bound in dimensions at least five, and local classification results above the classical CMC gap threshold. These results provide further evidence for the BMO conjecture.
Related
- Biharmonic Lagrangian H-umbilical submanifolds in complex space forms
- Biharmonic submanifolds in nonpositive space forms
- Biharmonic Lagrangian surfaces with constant Gaussian curvature in $2$-dimensional complex space forms
- Hypersurfaces with constant sectional curvature of a product of two space forms
- Biconservative hypersurfaces in space forms with harmonic curvature
BibTeX
@misc{fu2026biharmonic,
title = {{Biharmonic hypersurfaces in space forms}},
author = {Yu Fu and Min-Chun Hong and Dan Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04165},
eprint = {2610.04165},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04165},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Biharmonic hypersurfaces in space forms
AU - Yu Fu
AU - Min-Chun Hong
AU - Dan Yang
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.04165
UR - https://arxiv.org/abs/2610.04165
AB - We prove that every biharmonic hypersurface in a space form of nonpositive sectional curvature is minimal in arbitrary dimension. This settles the hypersurface case of the generalized Chen's conjecture in hyperbolic space and gives a unified treatment of nonpositive space forms. For biharmonic hypersurfaces in the unit sphere, we derive quantitative restrictions on every possible nonconstant-mean-curvature solution. In particular, we obtain pointwise criteria forcing constant mean curvature, a strict scalar-curvature bound in dimensions at least five, and local classification results above the classical CMC gap threshold. These results provide further evidence for the BMO conjecture.
ER -