15 problems

Gray's conjecture. Any homogeneous nearly Kähler manifold must be 3-symmetric.

Wolf–Gray conjecture. Every strict nearly Kähler $6$-manifold has the property asserted by the preceding Wolf–Gray theorem; namely, the result is true for strict nearly Kähler $6$-…

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.

Sawaki–Yamanoue's conjecture. Every $6$-dimensional strictly nearly Kähler manifold is a space of constant curvature.

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…

Consider the cohomogeneity one nearly Kähler metrics constructed by Foscolo and Haskins. Foscolo--Haskins' conjecture. The examples they construct are the only cohomogeneity one ne…

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…

Uniqueness conjecture. These are the only inhomogeneous complete cohomogeneity-one nearly Kähler structures in six dimensions.

Conjecture. The group $\operatorname{Diff}(M^6,[\omega])$ of diffeomorphisms preserving the conformal class of $\omega$ is a finite-dimensional Lie group. This concerns the automor…

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