Four-dimensional pseudo-umbilical biharmonic submanifolds in six-dimensional Riemannian manifolds of constant sectional curvature
Abstract
In 1989, Dimitrić showed that any pseudo-umbilical biharmonic submanifold in a Euclidean space is minimal when the dimension of the submanifold is different from four. However, the four-dimensional case has remained open for nearly four decades. This difficulty comes from the fact that the tangential part of the biharmonic equation vanishes identically in dimension four. Therefore, we consider biminimal submanifolds, whose defining equation generalizes the normal part of the biharmonic equation. In this paper, we show that any connected pseudo-umbilical 0-biminimal submanifold of codimension two and dimension at least three in a Riemannian manifold of constant sectional curvature c has constant mean curvature. Furthermore, when c is non-positive, we prove that it is minimal. As a corollary, we show that any four-dimensional pseudo-umbilical biharmonic submanifold in a six-dimensional Riemannian manifold of non-positive constant sectional curvature is minimal. These results also provide partial affirmative answers to Chen's conjecture, generalized Chen's conjecture for non-positive constant sectional curvature, and the Balmuş-Montaldo-Oniciuc conjecture.
Related
- Biharmonic Wintgen ideal submanifolds in Riemannian manifolds of constant sectional curvature
- Biharmonic Lagrangian H-umbilical submanifolds in complex space forms
- Pólya--Szegö Inequality on Submanifolds of Riemannian Manifolds with Nonnegative Curvature and Applications
- Computing Fréchet Means on Unknown Submanifolds of Riemannian Manifolds
- Rigidity of volume-minimizing free boundary hypersurfaces in high dimensional Riemannian manifolds
BibTeX
@misc{maeta2026fourdimensional,
title = {{Four-dimensional pseudo-umbilical biharmonic submanifolds in six-dimensional Riemannian manifolds of constant sectional curvature}},
author = {Shun Maeta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17136},
eprint = {2609.17136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17136},
note = {CC0},
}
RIS
TY - UNPB
TI - Four-dimensional pseudo-umbilical biharmonic submanifolds in six-dimensional Riemannian manifolds of constant sectional curvature
AU - Shun Maeta
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.17136
UR - https://arxiv.org/abs/2609.17136
AB - In 1989, Dimitrić showed that any pseudo-umbilical biharmonic submanifold in a Euclidean space is minimal when the dimension of the submanifold is different from four. However, the four-dimensional case has remained open for nearly four decades. This difficulty comes from the fact that the tangential part of the biharmonic equation vanishes identically in dimension four. Therefore, we consider biminimal submanifolds, whose defining equation generalizes the normal part of the biharmonic equation. In this paper, we show that any connected pseudo-umbilical 0-biminimal submanifold of codimension two and dimension at least three in a Riemannian manifold of constant sectional curvature c has constant mean curvature. Furthermore, when c is non-positive, we prove that it is minimal. As a corollary, we show that any four-dimensional pseudo-umbilical biharmonic submanifold in a six-dimensional Riemannian manifold of non-positive constant sectional curvature is minimal. These results also provide partial affirmative answers to Chen's conjecture, generalized Chen's conjecture for non-positive constant sectional curvature, and the Balmuş-Montaldo-Oniciuc conjecture.
ER -