10 problems

Classification conjecture. If a compact simply-connected $4$-dimensional Ricci soliton admits an invariant cohomogeneity one action, then it is isometric to a rescaling of one of t…

Let $d_1,d_2$ be positive integers and set $n=d_1+d_2+1$. Let $A(\alpha)$ and $\Omega(\omega)$ be the curves whose intersections correspond to…

Let $d_1,d_2$ be positive integers with $d_1+d_2+1=10$, and consider Einstein metrics on $S^{10}$ invariant under the usual cohomogeneity-one action of…

Let $m\geq 2$. Consider the cohomogeneity-one action with group triple … Its principal orbit is $\mathbb{S}^{4m+3}$, identified with the total space of the quaternionic Hopf fibrat…

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…

Uniqueness conjecture. Any such solution is the round sphere, up to diffeomorphism.

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.