Hamilton-Ivey estimates for the Ricci-Bourguignon flow with $\rho<0$

TL;DR

This paper establishes Hamilton-Ivey estimates for the Ricci-Bourguignon flow on 3D compact manifolds with negative , showing ancient solutions maintain nonnegative sectional curvature.

Contribution

It extends Hamilton-Ivey estimates to the Ricci-Bourguignon flow with <0 in three dimensions, providing new curvature bounds and properties of ancient solutions.

Findings

Hamilton-Ivey estimates are proven for the flow with <0

Ancient solutions have nonnegative sectional curvature

Results apply specifically to 3D compact manifolds

Abstract

In this paper, we prove Hamilton-Ivey estimates for the Ricci-Bourguignon flow on a compact manifold, with $n=3$ and $\rho<0$. As a consequence, we prove that compact ancient solutions have nonnegative sectional curvature for all negative $\rho$.