5 problems

Let $(X^N,g_+)$, with $N=n+1\ge 3$, be a smooth Poincaré–Einstein manifold with conformal infinity $(M^n,[h])$. Assume that $Y(M,[h])>0$, and let $(\overline X,[\bar g])$ denote it…

Regularity conjecture. The regularity of $\rho_s$ and $\bar{g}_s$ is given by the minimum of the background regularity of $\bar{g}$ and the fractional order $2s-n$.

Fractional positive mass conjecture. The constant term $A$ in this Green's function expansion is nonnegative, and is positive unless the underlying Poincaré–Einstein manifold is th…

Let $(X^{n+1},g_+)$, with $n\geq 5$, be a Poincaré-Einstein manifold with conformal infinity $(M,[g])$. Let $\mathcal{Y}_1(M,[g])$ and $\mathcal{Y}_2(M,[g])$ denote the first and s…

Let $\gamma \in (0,1)$, $n > 2\gamma$, and let $(X^{n+1},g^+)$ be Poincaré–Einstein. Suppose that $\Lambda^{\gamma}(M,[\hat{h}])>0$ and that either $(M^n,[\hat{h}])$ is locally con…