20 problems

Green function rigidity conjecture. Then $(M,g)$ is a round sphere.

Let $n$ be the dimension of the unit sphere $athbb{S}^n$ and let $L_{2k}$ be the $k$-th GJMS operator … Let $0<l<k<n/2$. Higher-order GJMS Sobolev inequality conjecture. The follow…

Let $k,m\in\mathbb{N}$ and let $N(k,m)$ be the dimension threshold appearing in the higher-order GJMS geometric Aubin set conjecture for conformal manifolds virtually conformal to…

The higher-order GJMS geometric Aubin set conjecture. The set $\mathscr{A}_{n,2k}$ is a geometric Aubin set for $L_{2k}$.

Green-function formula conjecture. In critical dimension, the Green function of $P_n$ should be

Let $n\geq 2$, let $\mathbb{S}^n$ be the round sphere, let $P_{\mathbb{S}^n}$ denote its GJMS operator, let $Q_{g_{\mathbb{S}^n}}=(n-1)!/2$ be its constant $Q$-curvature, and let…

Let $M$ be a compact orientable Riemannian manifold of even dimension $n$, let ${\mathcal T}_M=({\mathcal M}/{\mathcal P})/{\mathcal D}_0$ be its Teichmüller space of conformal str…

Let $M$ be a compact orientable Riemannian manifold of even dimension $n$, let ${\mathcal T}_M=({\mathcal M}/{\mathcal P})/{\mathcal D}_0$ be its Teichmüller space of conformal str…

Let $M$ be a compact orientable Riemannian manifold, let ${\mathcal T}_M=({\mathcal M}/{\mathcal P})/{\mathcal D}_0$ be its Teichmüller space of conformal structures, and let $P_k$…

Let $M$ be a compact orientable Riemannian manifold, let ${\mathcal T}_M=({\mathcal M}/{\mathcal P})/{\mathcal D}_0$ be its Teichmüller space of conformal structures, and let $P_k$…

Let … be the self-adjoint operator formed from compositions of GJMS-operators $P_{2I}$, where the sum runs over compositions $I$ of $N$ and $m_I$ are the associated coefficients. J…

Let $I=(I_1,\dots,I_r)$ be a composition of $N$, let $I_{\mathrm{last}}=I_r$, and define $P_{2I}=P_{2I_1}\circ\cdots\circ P_{2I_r}$ with multiplicities $m_I$ as above. Let…

Let $I=(I_1,\dots,I_r)$ be a composition of $N$, let $P_{2I}=P_{2I_1}\circ\cdots\circ P_{2I_r}$, and define … where … For a locally conformally flat manifold of dimension $n\geq 3$…

Let $N\geq1$ satisfy $2N\leq n$ when $n$ is even. On any Riemannian manifold $(M,g)$ of dimension $n\geq3$, let $D_{2},D_{4},\ldots$ be the symmetric bilinear forms defined by … wh…

Let $P_{2N}^{res}(\lambda)$ be the recursively defined residue-family polynomial of degree $N-1$ built from the GJMS-operators, and let ${\mathcal M}_{2N}$ be the associated operat…

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…

Let $P_{2N}$ be the GJMS-operator of order $2N$ on a locally conformally flat manifold, and let $P_{2N}^0$ denote its non-constant part. Let ${\mathcal P}_{2N}$ be a linear combina…

Let $(M,g)$ be a $pp$-wave, and let $P_{2k}$ denote its GJMS operators. Since the scalar curvature of a $pp$-wave vanishes, the Laplacian $44_g$ is conformally covariant; calculati…

Promotion conjecture. The operators $P_n$ are the only longest arrows that promote.