吉利亚-曼泰加扎流与里奇流的二阶偏离
Second-Order Departure of the Gigli--Mantegazza Flow from Ricci Flow
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中文总结 AI 辅助
该研究推导了Gigli--Mantegazza流的二阶展开式，证明其与里奇流的二阶偏差由曲率决定，且二者的Gromov--Hausdorff距离为$O(t^2)$，圆球面实例验证了该估计的精确性。
中文摘要 AI 辅助
对于闭连通黎曼流形$(M,g)$，Gigli--Mantegazza构造在热核嵌入$x\mapsto p_t(x,\cdot)\\,d\operatorname{vol}_g$下拉回二次Wasserstein度量，所得族$\widetilde{g}_t$在$t$的一阶项上与里奇流一致，但通常二阶项不一致。本文证明$\widetilde{g}_t =g-2t\operatorname{Ric}_g +t^2\left(-\Delta\operatorname{Ric}_g +2\operatorname{Ric}_g^2-\frac{2}{3}\mathcal{Q}_g\right) +O_{C^0}(t^3)$，其中$\mathcal{Q}_g$是全曲率张量的二次项。项$-\Delta\operatorname{Ric}_g$也出现在里奇流的二阶展开中，故偏差逐点依赖曲率且呈二次关系。特别地，在非平坦的里奇平坦度量处，里奇流是平稳的而$\widetilde{g}_t$并非如此。Gromov--Hausdorff距离在Gigli--Mantegazza度量与里奇流度量间为$O(t^2)$，而圆球面表明该估计是精确的。
英文摘要
For a closed connected Riemannian manifold $(M,g)$, the Gigli--Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding $x\mapsto p_t(x,\cdot)\,d\operatorname{vol}_g$. The resulting family $\widetilde{g}_t$ agrees with Ricci flow to first order in $t$, but in general not to second order. We prove that $$ \widetilde{g}_t =g-2t\operatorname{Ric}_g +t^2\left(-Δ\operatorname{Ric}_g +2\operatorname{Ric}_g^2-\frac{2}{3}\mathcal{Q}_g\right) +O_{C^0}(t^3), $$ where $\mathcal{Q}_g$ is quadratic in the full curvature tensor. The term $-Δ\operatorname{Ric}_g$ 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 $\widetilde{g}_t$ 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.
