Singularities of Connection Ricci Flow and Ricci Harmonic Flow

TL;DR

This paper investigates singularities in connection Ricci flow and Ricci harmonic flow, defining singularity types, models, and convergence, and explores properties of ancient solutions using curvature quantities and monotonicity formulas.

Contribution

It introduces new curvature-based definitions of singularities, establishes convergence results, and links finite-time singularities with Ricci harmonic solitons, advancing understanding of these flows.

Findings

Defined three types of singularities and their models.

Proved convergence of singularity models.

Linked finite-time singularities with shrinking Ricci harmonic solitons.

Abstract

In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities and their corresponding singularity models, and then prove the convergence. In addition, for Ricci harmonic flow, we use the monotonicity of functional $\nu_\alpha$ to show the connection between finite-time singularity and shrinking Ricci harmonic soliton. At last, we explore the property of ancient solutions for Ricci harmonic flow.