10 problems

Compact nearly Kähler conjecture. Every compact nearly Kähler manifold is a 3-symmetric space.

Gray and Wolf's conjecture. Every nearly Kähler homogeneous manifold is a 3-symmetric space equipped with its canonical almost complex structure.

Gray–Wolf conjecture. Every homogeneous strictly nearly Kähler space is a naturally reductive 3-symmetric space equipped with its canonical almost-complex structure.

Existence and uniqueness conjecture. There exists an isometric immersion $f:M\to(\mathbb{C}P^3,g,J)$ and a vector bundle isometry $\varphi:\mathcal{E}\to T^\perp M$ such that…

Let $(\overline{X},\omega_{CS},\rho_{CS})$ be a nearly Kähler conifold with a singularity at $x_0$ modelled on the Calabi--Yau cone $(C=C(L),\omega_C,\rho_C)$. An obstruction conje…

A compact nearly Kähler manifold is cohomogeneity one if it is equipped with an isometric action of a compact Lie group whose generic orbits have codimension one. Foscolo–Haskins c…

Zero-count conjecture. The count satisfies: 1. $\mathcal{C}(a)$ and $\mathcal{C}(b)$ are decreasing in $a$ and $b$, respectively. 2. For $a>0$ sufficiently small,…

Monotonicity conjecture. The functions $V(\Psi_a)$ and $V(\Psi_b)$ are strictly increasing in $a$ and $b$, respectively.

Classification conjecture. The Main Theorem yields all inhomogeneous complete cohomogeneity-one nearly Kähler structures. In particular, $S^2\times S^4$ admits no cohomogeneity-one…

Butruille's conjecture. Every compact nearly Kähler manifold is a 3-symmetric space.