9 problems

Let $\mathbb S^{l}(\frac{1}{2})\times\mathbb S^{l}(\frac{1}{2})$ be the generalised Clifford torus, viewed as a submanifold of $\mathbb S^{2l+2}$, and consider the associated bihar…

Let $M$ be a biharmonic submanifold of a sphere. BMO conjecture. The submanifold $M$ has constant mean curvature. The source presents this as one of the well-known open conjectures…

Let $M$ be a biharmonic submanifold of a non-positively curved Riemannian manifold. Generalized Chen's conjecture. The submanifold $M$ is minimal. Ou and Tang constructed counterex…

Let $M$ be a biharmonic submanifold of a Euclidean space. Chen's conjecture. Every such submanifold is minimal. Chen, Ishikawa, and Jiang proved the claim for biharmonic surfaces i…

Let $M$ be a compact, simply-connected, nonspin four-dimensional almost complex manifold. An energy-minimizing biharmonic almost complex structure on $M$ determines a homotopy clas…

Let $(g,J)$ be a compatible almost Hermitian structure, and define … A biharmonic almost complex structure is a zero of $B$. Transversality conjecture. The map $B$ is transverse to…

Let $M$ be a complete submanifold of Euclidean space $\mathbb{E}^n$. An isometric immersion $\phi:(M,g)\rightarrow\mathbb{E}^n$ is biharmonic when its bitension field satisfies…

Let $M$ be a $(2n+1)$-dimensional semi-Riemannian space form $M_{1}^{2n+1}(c)$ of index $1$, and let $B$ be a $2n$-dimensional Riemannian manifold. A proper biharmonic semi-Riemann…

Complete Chen conjecture. Any complete biharmonic submanifold in $\mathbb{E}^N$ is minimal.