Evolution and Monotonicity of Geometric Constants under Extended Ricci Flows with Variable Coupling Parameters

Abstract

This paper explores the evolution and monotonicity of geometric constants within the framework of extended Ricci flows, incorporating variable coupling parameters. Building on Hamiltons foundational Ricci flow and subsequent extensions by List (2008), we introduce modifications to the extended Ricci flow by varying parameters that affect the interaction between the metric and scalar fields. Specifically, we modify the coefficients in the evolution equations governing geometric constants, thereby introducing new degrees of freedom in the analysis. The primary contributions include deriving evolution formulas for the modified geometric constant lambda under the extended and normalized extended Ricci flows, and proving conditions under which monotonicity is maintained.