Scalar curvature, entropy, and generalized Ricci flow

TL;DR

This paper introduces new monotonicity formulas for scalar curvature and entropy in generalized Ricci flow, involving a dilaton field and dual energy functionals, with implications for convexity and pseudolocality in Ricci flow.

Contribution

It develops novel weighted scalar curvature monotonicity formulas and dual entropy functionals for generalized Ricci flow, extending Perelman’s framework with a dilaton field.

Findings

Derived scalar curvature monotonicity formulas for generalized Ricci flow.

Established dual energy and entropy monotonicity formulas involving a dilaton field.

Obtained new convex Nash entropies and pseudolocality principles for Ricci flow.

Abstract

We derive a family of weighted scalar curvature monotonicity formulas for generalized Ricci flow, involving an auxiliary dilaton field evolving by a certain reaction-diffusion equation motivated by renormalization group flow. These scalar curvature monotonicities are dual to a new family of Perelman-type energy and entropy monotonicity formulas by coupling to a solution of the associated weighted conjugate heat equation. In the setting of Ricci flow, we further obtain a new family of convex Nash entropies and pseudolocality principles.