Strong uniqueness of tangent flows at cylindrical singularities in Ricci flow

TL;DR

This paper proves the strong uniqueness of cylindrical tangent flows at singularities in Ricci flow by establishing a Lojasiewicz inequality for the pointed -entropy, ensuring convergence to cylindrical models near singularities.

Contribution

It introduces a Lojasiewicz inequality for the -entropy in Ricci flow and applies it to demonstrate the strong uniqueness of cylindrical tangent flows at singularities.

Findings

Established a Lojasiewicz inequality for -entropy near cylindrical geometries.

Proved convergence of Ricci flow to cylindrical models at singularities.

Demonstrated strong uniqueness of tangent flows at the first singular time.

Abstract

In this paper, we establish a Lojasiewicz inequality for the pointed $\mathcal{W}$-entropy in the Ricci flow, under the assumption that the geometry near the base point is close to a standard cylinder $\mathbb{R}^k \times S^{n-k}$ or the quotient thereof. As an application, we prove the strong uniqueness of the cylindrical tangent flow at the first singular time of the Ricci flow. Specifically, we show that the modified Ricci flow near the singularity converges to the cylindrical model under a fixed gauge.