A sharp diameter-mass inequality on Einstein manifolds and ADM mass

Let $(M^n,g),n\ge 4$ be a close Einstein manifold with positive scalar curvature and $G$ be the Green function of the conformal Laplacian. The ADM mass of the asymptotically flat metric $\widehat{g}:=G^{4\mathrm{/}(n-2)}g$ is an important geometric invariant. When either $n=5$ or $n\ge 6$ with Weyl vanishing condition, we prove a mass formula, generalizing Gursky-Malchiodi’s formula in dimension $4$. An important step is an extension of Viaclovsky’s Green function expansion in higher dimensions. Next we prove a sharp diameter-mass inequality on Einstein manifolds. Finally we present two applications. First, we prove a mass gap theorem for Einstein manifolds. Second, for Collins-Székelyhidi’s family of Sasaki-Einstein metrics $g_i$ on ${\mathbb{S}}^5$ with volume going to zero, the sequence of the masses $m(\widehat{g_i})$ blows up uniformly as $i\to \mathrm{\infty }$.