Strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow

TL;DR

This paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow by establishing a Lojasiewicz inequality and analyzing tangent flows near generalized cylinders.

Contribution

It extends previous results to generalized cylinders, proving strong uniqueness of tangent flows and rectifiability of the singular set in Ricci flow.

Findings

Proves strong uniqueness of generalized cylindrical tangent flows.

Establishes a Lojasiewicz inequality for the pointed ntropies.

Shows rectifiability of the singular set at cylindrical points.

Abstract

In this paper, we extend the results of \cite{fang2025strong, fang2025singular} to generalized cylinders. More precisely, we establish a Lojasiewicz inequality for the pointed $\mathcal{W}$-entropy in Ricci flow under the assumption that the geometry near the base point is close to a generalized cylinder $\mathbb{R}^k \times N^{n-k}$, where $N$ is an Einstein manifold with obstruction of order three satisfying a suitable spectral condition. As an application, we prove the strong uniqueness of generalized cylindrical tangent flows. Furthermore, we show that the subset $\mathcal{S}^k_{\mathrm{qc}}(N)\subset \mathcal{S}^k$, consisting of points at which some tangent flow is given by $\mathbb{R}^k \times N^{n-k}$ or its quotient, is horizontally parabolic $k$-rectifiable.