里奇流热核的费舍尔度量
The Fisher Metric of the Ricci Flow Heat Kernel
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中文总结 AI 辅助
该研究引入里奇流共轭热核的费舍尔度量，证明其单调性、与纳什熵的关联等性质，推导相关不等式、收缩公式等推论，并给出Bamler正则性定理的应用判据。
中文摘要 AI 辅助
我们引入并研究与里奇流\f(M^n,g_t\f)的共轭热核相关联的费舍尔信息度量\f(g^F_\tau\f)。该张量在固定尺度下测量当基点移动时，带点热核测度的变化情况。我们证明\f(g^F_\tau\f)随尺度单调变化，且满足\f(g^F_\tau\le g_t\f)。我们将其迹与带点纳什熵关联起来，并证明费舍尔亏格\f(g_t-g^F_\tau\f)的矩阵平方恒等式，该恒等式给出了等号情形的刚性定理：在闭连通流上，每个正尺度都有严格不等式\f(0<g^F_\tau<g_t\f)；在完备情形下，等号会迫使欧氏分裂。我们还发展了这一观点的若干推论，包括热半群的尖锐反向庞加莱不等式、沿共轭热流的\f(\varphi\f)-散度收缩公式，以及从费舍尔自同态的大特征值出发的热核分裂映射的典范构造。作为应用，我们将接近0的带点纳什熵与可比尺度下的小额费舍尔亏格关联起来，并得到了应用Bamler的\f(\varepsilon\f)-正则性定理的余维一费舍尔度量判据。
英文摘要
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 \(φ\)-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.
