11 problems

A submanifold of Euclidean space is biharmonic when its inclusion map is a biharmonic map, equivalently a critical point of the bienergy functional. It is minimal when its mean cur…

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

Let … be the distinguished family of proper biharmonic maps, and let $g(k)$ denote the number of mixed Fourier modes for which the corresponding explicit Fourier eigenvalue vanishe…

Let a biharmonic hypersurface be a hypersurface in a sphere whose inclusion map is biharmonic, and let its mean curvature be the trace-normalized mean curvature function. Constant-…

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…

For each positive integer $k$, let $\varphi_k$ denote the proper biharmonic map considered in the paper, and let ${\rm Nullity}(\varphi_k)$ denote its nullity, the dimension of the…

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.

Theorem 1.1 concerns a sequence of approximate biharmonic maps into a sphere in dimension four and asserts an energy identity; let $N$ be a target manifold. Extension conjecture. T…