Ancient solutions of Ricci flow with Type I curvature growth

TL;DR

This paper classifies certain ancient solutions to the Ricci flow with Type I curvature growth, showing they split Euclidean factors or are locally symmetric under specific conditions.

Contribution

It proves that noncompact, noncollapsed Type I ancient solutions with nonnegative sectional curvature split Euclidean factors, and solutions with weakly PIC2 are locally symmetric.

Findings

Noncompact, noncollapsed Type I solutions with nonnegative sectional curvature split Euclidean factors.

Solutions with weakly PIC2 are necessarily locally symmetric.

Provides classification results for ancient Ricci flow solutions with Type I curvature growth.

Abstract

Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type~I curvature growth. We show that a $\kappa$-noncollapsed Type~I ancient solution which is noncompact and has nonnegative sectional curvature necessarily splits at least one Euclidean factor. It follows that a $\kappa$-noncollapsed Type~I ancient solution which is weakly PIC2 is a locally symmetric space.