Locally conformally flat ancient Ricci flows

TL;DR

This paper proves that locally conformally flat ancient Ricci flows are rotationally symmetric and that such Ricci solitons with nonnegative curvature are gradient solitons across shrinking, steady, and expanding cases.

Contribution

It establishes the rotational symmetry of locally conformally flat ancient Ricci flows and characterizes locally conformally flat Ricci solitons as gradient solitons under nonnegative curvature.

Findings

Ancient solutions are rotationally symmetric.

Locally conformally flat Ricci solitons are gradient solitons with nonnegative curvature.

Results unify the understanding of symmetry and soliton structure in Ricci flow.

Abstract

We show that any locally conformally flat ancient solution to the Ricci flow must be rotationally symmetric. As a by-product, we prove that any locally conformally flat Ricci soliton is a gradient soliton in the shrinking and steady cases as well as in the expanding case, provided the soliton has nonnegative curvature.