Positive mass theorem for the fractional Yamabe problem

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 conformally flat or $n=2$. Let $G(\cdot,y)$ be the Green's function associated with the fractional conformal problem, and let $d_{\bar g}$ denote the distance induced by the compactified metric $\bar g$. Positive mass theorem. For any $x\in\overline{X}$ sufficiently near $y\in M$, the Green's function has an expansion

$$G(x,y)=g_{n,\gamma}\,d_{\bar g}(x,y)^{-(n-2\gamma)}+A+\Psi(d_{\bar g}(x,y)),\qquad A\geq 0,$$

where $g_{n,\gamma}>0$ is the constant appearing in the model bubble. Here $\Psi$ is defined on a small closed neighborhood $\mathcal{N}\subset\overline{\mathbb{R}^{n+1}_+}$ of $0$, satisfies $\Psi(0)=0$, and for some $\vartheta_1\in(0,1)$ obeys

$$\|\Psi\|_{C^{\vartheta_1}(\mathcal{N})}+\|\nabla_{\bar x}\Psi\|_{C^{\vartheta_1}(\mathcal{N})}+\left\|x_{n+1}^{1-2\gamma}\frac{\partial\Psi}{\partial x_{n+1}}\right\|_{C^{\vartheta_1}(\mathcal{N})}\leq C.$$

Moreover, $A=0$ if and only if $(X^{n+1},\bar g)$ is conformally diffeomorphic to the standard unit ball $\mathbb{B}^{n+1}$, denoted $(X^{n+1},\bar g)\simeq\mathbb{B}^{n+1}$. This is a fractional analogue of the positive mass theorem: nonnegativity of the constant term in the Green's-function expansion, together with its rigidity case, is expected to constrain the geometry of Poincaré–Einstein manifolds.