Second-Order Departure of the Gigli--Mantegazza Flow from Ricci Flow

For a closed connected Riemannian manifold $(M,g)$, the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding $x\mapsto p_t(x,\cdot )d{\mathrm{vol}}_g$. The resulting family ${\tilde{g}}_t$ agrees with Ricci flow to first order in $t$, but in general not to second order. We prove that $${\tilde{g}}_t=g-2t{\mathrm{Ric}}_g+t^2(-\mathrm{\Delta }{\mathrm{Ric}}_g+2{\mathrm{Ric}}_g^2-\frac{2}{3}{\mathcal{Q}}_g)+O_{C^0}(t^3),$$ where ${\mathcal{Q}}_g$ is quadratic in the full curvature tensor. The term $-\mathrm{\Delta }{\mathrm{Ric}}_g$ also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while ${\tilde{g}}_t$ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is $O(t^2)$, and round spheres show that this estimate is sharp.