16 problems

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. Nonnegative curvature implication conjecture. If … then … This is motivated by results for conformally flat four-m…

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold. Noncompact Bonnet–Myers conjecture. If … then $M^4$ is compact. This extends the corresponding proposition from co…

Positivity conjecture for lower-order Q-curvatures. If $Q^{(2m)}_g$ is positive and has a slow decay barrier with rate $-2m<s\leq 0$ at infinity, then, for every $1\leq k\leq m-1$,

Dual higher-order Bol inequality conjecture. One should have

Higher-order Bol inequality conjecture. One should have

Conjectural decomposition. If $n$ is even, then ${\bf Q}_n(g)$ is a linear combination of $Q_n(h)$, local conformal invariants of the embedding $M\hookrightarrow X$, and a total di…

Extremal-parameter conjecture. For the problem $Q_{\lambda}$, the extremal parameter is

Extremal-parameter conjecture. The extreme value for $P_{\lambda}$ is

Q-curvature volume comparison conjecture. If

Vanishing conjecture for residue Q-curvatures. For every $N\ge 1$,

Fefferman–Hirachi conjecture. There exists a contact form $\theta$ in the conformal contact class

Let $n\geq 3$, and let $(M^n,\bar{g})$ be a connected closed flat Riemannian manifold. A metric has pointwise positive Q-curvature if its Q-curvature is positive at every point. Ri…

Divergence conjecture. On CR manifolds, the $Q$-curvature is a divergence: there exist form-valued local invariants $T_a$ and $T_{\overline{a}}$ of $\theta$ such that

Let … where $\Lambda_{2N}=Q_{2N}-{\mathcal Q}_{2N}$. Let $v(r)$ be the generating function defined by the volume coefficients. Square-root conjecture. … This reformulates the recur…

Eighth-order recursive formula.

Let $(M,g)$ be a locally conformally flat Riemannian manifold of dimension $n\geq3$. For $N\geq1$, let ${\mathcal M}_{2N}$ denote the operator defined from the recursive constructi…