The Fisher Metric of the Ricci Flow Heat Kernel

Abstract: We introduce and study a Fisher information metric (g^F_τ) associated to the conjugate heat kernel of a Ricci flow ((M^n,g_t)). This tensor measures, at a fixed scale, how the pointed heat-kernel measure changes when the base point is moved. We prove that (g^F_τ) is monotone in scale and satisfies (g^F_τ\le g_t). We relate its trace to the pointed Nash entropy and prove a matrix square identity for the Fisher defect (g_t-g^F_τ.) This identity gives a rigidity theorem for the equality case; on closed connected flows one has the strict inequalities (0<g^F_τ<g_t) at every positive scale, while in the complete case equality forces a Euclidean splitting. We also develop several consequences of this point of view. These include a sharp reverse Poincaré inequality for the heat semigroup, a contraction formula for (\varphi)-divergences along conjugate heat flow, and a canonical construction of heat-kernel splitting maps from large eigenvalues of the Fisher endomorphism. As applications, we relate pointed Nash entropy close to (0) to small Fisher deficit at comparable scales, and we obtain a codimension-one Fisher-metric criterion for applying Bamler's (\varepsilon)-regularity theorem.