Perelman's entropy and heat kernel bounds on RCD spaces
Camillo Brena
TL;DR
The paper extends Perelman’s W-entropy to non-smooth $RCD(K,N)$ spaces, establishing a rigorous time-derivative formula, monotonicity, and a rigidity dichotomy. It develops Eulerian heat-flow controls and sharp heat-kernel bounds on $RCD(K,N)$ spaces, including a careful handling of potential collapse and non-collapsed cases via a cutoff-entropy framework. The derivative formulas for both cutoff and full entropy yield monotonicity and, under equality, strong rigidity: the space is either conical or modelled by a Euclidean-type geometry. These results provide tools for analyzing singular and Ricci-limit spaces, with potential applications to the geometric structure of non-smooth spaces and rectifiability properties of singular sets.
Abstract
We study Perelman's W-entropy functional on finite-dimensional RCD spaces, a synthetic generalization of spaces with Bakry-Émery Ricci curvature bounded from below. We rigorously justify the formula for the time derivative of the W-entropy and derive its monotonicity and rigidity properties. Additionally, we establish bounds for solutions of the heat equation, which are of independent interest.