Biharmonic surfaces with flat normal bundle in the four-dimensional sphere
Abstract
We prove that a proper biharmonic surface with flat normal bundle in the four-dimensional unit sphere 𝕊4 lies minimally in a small hypersphere of radius 1/√2. Since the surfaces with constant mean curvature and those with parallel normalized mean curvature vector field are already understood, the core of the paper is the non-existence of biharmonic surfaces with flat normal bundle in 𝕊4 whose mean curvature is not constant and whose normalized mean curvature vector field is not parallel. We first show, adapting an argument of Fu, Hong and Yang, that the gradient of the mean curvature is a principal direction. Then the problem reduces to a first order system of ordinary differential equations with a constraint, and we show that this system has no admissible solutions by an elimination argument.
Related
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- Codimension Reduction of Minimal Submanifolds with Flat Normal Bundle in Spheres
- Compact Kähler surfaces with semipositive anticanonical bundle
- Normal Bundle Splitting Strata of Rational Curves in Toric Varieties
BibTeX
@misc{andronic2026biharmonic,
title = {{Biharmonic surfaces with flat normal bundle in the four-dimensional sphere}},
author = {Ştefan Andronic and Stefano Montaldo and Cezar Oniciuc and Antonio Sanna},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08066},
eprint = {2611.08066},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08066},
}
RIS
TY - UNPB
TI - Biharmonic surfaces with flat normal bundle in the four-dimensional sphere
AU - Ştefan Andronic
AU - Stefano Montaldo
AU - Cezar Oniciuc
AU - Antonio Sanna
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08066
UR - https://arxiv.org/abs/2611.08066
AB - We prove that a proper biharmonic surface with flat normal bundle in the four-dimensional unit sphere 𝕊^4 lies minimally in a small hypersphere of radius 1/√2. Since the surfaces with constant mean curvature and those with parallel normalized mean curvature vector field are already understood, the core of the paper is the non-existence of biharmonic surfaces with flat normal bundle in 𝕊^4 whose mean curvature is not constant and whose normalized mean curvature vector field is not parallel. We first show, adapting an argument of Fu, Hong and Yang, that the gradient of the mean curvature is a principal direction. Then the problem reduces to a first order system of ordinary differential equations with a constraint, and we show that this system has no admissible solutions by an elimination argument.
ER -