44 problems

Let $(M^4,g)$ be a complete four-dimensional Riemannian manifold whose scalar curvature is bounded below by a positive constant and whose quotient satisfies $Q_g/R_g\geq k$ for som…

Let $M$ be a CR manifold with a pseudo-Einstein contact form, and let a pseudo-hermitian invariant be defined for such contact forms. Suppose that its integral gives a global secon…

Let $(M^3,J,\theta)$ be a closed three-dimensional pseudohermitian manifold, and let $[\theta]$ denote its contact class. A contact form in this class is said to have vanishing CR…

Green-function coefficient conjecture. At this point,

A natural density $U$ is a scalar density constructed naturally from the metric and its derivatives, and a Polyakov-type formula has a local part consisting of terms integrated aga…

Let $P_k$ be the differential operator associated to a parallel standard tractor on a conformally Einstein manifold, acting on tractor fields of weight $k-n/2$, and let $\Box_k^0$…

Let $M$ be a strictly pseudoconvex CR manifold of dimension $2N-1$, let $\theta$ be a contact form, and let $S$ be a scalar pseudohermitian invariant satisfying the CR pluriharmoni…

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. The lower-order $Q$-curvature $Q_g^{(2k)}$ should be positive.

Let $g=e^{2u}|dx|^2$ be a smooth complete normal metric on $\mathbb{R}^n$ with finite total $Q$-curvature, where $n\geq 3$. Let $I_g$ denote the isoperimetric ratio, $Q^{(n)}_g$ th…

Let $D=2m$ be an arbitrary even dimension, and consider the conformal obstruction tensor together with its four defining conditions from Graham and Hirachi. Uniqueness conjecture.…

In even dimension $D=2m$, consider conformal gravity actions whose Euler–Lagrange equations admit every Einstein metric as a solution. Uniqueness conjecture. There is only one such…

Lower-order Q-curvature conjecture. One should have

Isoperimetric-ratio-free growth conjecture. The condition $I_g>0$ should be unnecessary for the conclusion that, for every integer $1\leq k\leq\frac{n-2}{2}$ and every fixed point…

Let $J_{\alpha}(u)$ be the functional associated with a $Q$-curvature-type equation, and let $C(\alpha)$ denote a lower-bound constant for this functional. Gui-Li-Wei-Ye's conjectu…

Groupoid-substructure conjecture. Varying $Q_{2n}$ gives a linear operator in $d=2n+2$ which can act as a definition of a groupoid substructure of the Weyl group.

Linearity conjecture. The shift-operator coming from the transformation of the $2n$-th $Q$-curvature generates a linear differential constraint in $d=2n+2$.

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 $1\leq k\leq m-1$,

Andrade–Piccione–Wei conjecture. Under these hypotheses, there is a positive minimizer for $Y_{Q_6}(M^n,[g])$. This concerns existence of a minimizing metric for the sixth-order Ya…

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