5 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…

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…