Evolution of Functionals Under Extended Ricci Flow

TL;DR

This paper studies how specific scalar functionals evolve under the extended Ricci flow on compact manifolds, providing explicit formulas without restrictive assumptions, advancing understanding of geometric and scalar interactions.

Contribution

It derives explicit evolution formulas for functionals involving powers of a scalar quantity under the extended Ricci flow, without assumptions on the manifold or scalar fields.

Findings

Explicit formulas for the time derivatives of scalar functionals.

Insights into the interplay between geometry and scalar functions.

General results applicable to various scalar powers and flows.

Abstract

In this paper, we investigate the evolution of certain functionals involving higher powers of a scalar quantity $F$ under Bernard List's extended Ricci flow on a compact Riemannian manifold. By deriving explicit expressions for the time derivative of integrals of the form $\int_M F^n \cdot \frac{\partial F}{\partial t} \, d\mu$ for various powers $n$, we explore the intricate interplay between geometric quantities and scalar functions without making any assumptions about the manifold, the scalar field $\Phi$, or the function $u$.