A complete proof of Hamilton's conjecture

TL;DR

This paper provides a complete proof of Hamilton's conjecture, demonstrating that certain complete 3-manifolds with positive scalar curvature and Ricci pinching are necessarily compact, using advanced Ricci flow techniques.

Contribution

The paper offers the first full proof of Hamilton's conjecture for 3-manifolds with Ricci pinching, employing novel estimates and flow analysis to exclude noncompact cases.

Findings

Proves that such 3-manifolds are compact under given conditions.

Develops pinching-decaying estimates to prevent local collapse.

Constructs Ricci expanders using monotonicity formulas.

Abstract

In this paper, we give the full proof of a conjecture of R.Hamilton that for $(M^3, g)$ being a complete Riemannian 3-manifold with bounded curvature and with the Ricci pinching condition $Rc\geq \ep R g$, where $R>0$ is the positive scalar curvature and $\ep>0$ is a uniform constant, $M^3$ is compact. One of the key ingredients to exclude the local collapse in singularities of the Ricci flow is the use of pinching-decaying estimate. The other important part of our argument is to role out the Type III singularity complete noncompact Ricci flow with positive Ricci pinching condition. We get this goal by obtaining an Ricci expander based on the monotonicity formula of weighted reduced volume.