Curvature Blow-up in Doubly-warped Product Metrics Evolving by Ricci Flow

TL;DR

This paper constructs examples of Ricci flow on high-dimensional manifolds that develop type II singularities with curvature blowing up at rates faster than classical models, specifically on doubly-warped product spaces.

Contribution

It provides a rigorous construction of local conical singularities with arbitrarily high curvature blow-up rates in Ricci flow on doubly-warped product manifolds.

Findings

Existence of Ricci flow solutions with curvature blow-up rate (T-t)^{-k} for any k > 1.

Singularities modeled on Ricci-flat cones.

Construction of type II singularities in high-dimensional warped products.

Abstract

For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\| Rm \|_{\infty} (t) \gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales.