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

This article explores the conceptual and mathematical parallels between Perelman’s entropy functional in geometric analysis and Bianconi’s action in entropic quantum gravity. Historically, Perelman’s work on the Ricci flow revolutionized geometric analysis by providing tools to resolve the Poincar´e conjecture, while Bianconi’s entropic 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 formalism bridges geometric flows and image processing, offering insights into emergent spacetime dynamics and machine learning. This synthesis extends the legacy of variational calculus in physics, connecting Thurston’s geometrization program with modern data-driven optimization paradigms