On the Entropic Connection Between Perelman's Functional and Bianconi's Action

This article explores the conceptual and mathematical parallels between Perelman’sentropy functional in geometric analysis and Bianconi’s action in entropic quantumgravity. Historically, Perelman’s work on the Ricci flow revolutionized geometric analysis by providing tools to resolve the Poincar´e conjecture, while Bianconi’sentropic approaches emerged from information-theoretic formulations of quantum gravity. By establishing a correspondence through variable substitution and entropic principles, we demonstrate that both frameworks describe gradient flows aimed at minimizing geometric and informational "disorder"respectively. The unified formalismbridges geometric flows and image processing, offering insights into emergent spacetimedynamics and machine learning. This synthesis extends the legacy of variational calculus in physics, connecting Thurston’s geometrization program with modern data-driven optimization paradigms.本文探讨了几何分析中的佩雷尔曼熵泛函（Perelman’s entropy functional）与熵量子引力（entropic quantum gravity）领域比安科作用量（Bianconi’s action）之间的概念与数学类比。从历史脉络而言，佩雷尔曼在里奇流（Ricci flow）方向的研究通过提供解决庞加莱猜想（Poincaré conjecture）的工具，彻底革新了几何分析领域；而比安科的熵方法则源自量子引力的信息论表述。通过变量替换与熵原理建立对应关系后，我们证明两类框架均描述了旨在分别最小化几何与信息“无序度”的梯度流。该统一形式体系架起了几何流与图像处理之间的桥梁，为涌现时空动力学与机器学习领域提供了全新研究视角。这一综合研究延续了物理学中变分法的学术传统，将瑟斯顿几何化纲领（Thurston’s geometrization program）与现代数据驱动优化范式相联结。