Quantitative Rigidity of Ricci Nonnegative Manifolds via the Regular Green Constant

For a Ricci non-negative non-parabolic manifold $(M^{n\ge3},g)$, assuming the minimal normalized Green function admits the asymptotic expansion $G=r^{2-n}+H_p+O_1(r)$, we prove the scaling invariant defect $H_pR^{n-2}$ controls the geometry of the geodesic ball quantitatively, $\frac{1}{R}d_{GH}(B_p(R),B_0^{\mathbb{R}^n}(R))\le C_n \left(H_p R^{n-2}\right)^{\frac{1}{88(n+1)(4n+1)}}$.