On type-II singularities in Ricci flow on $\mathbb{R}^{N}$

TL;DR

This paper constructs a family of Ricci flow solutions on Euclidean space that develop type-II singularities with controlled blow-up rates, converging locally to known solitons, revealing new behaviors of singularity formation.

Contribution

It introduces a new class of solutions with prescribed singularity rates and demonstrates convergence to Bryant and cylinder solitons, advancing understanding of Ricci flow singularities.

Findings

Solutions encounter finite-time singularities with arbitrary slow formation.

Blow-ups near the origin converge to Bryant soliton.

Blow-ups at infinity converge to shrinking cylinder soliton.

Abstract

In each dimension $N\geq 3$ and for each real number $\lambda\geq 1$, we construct a family of complete rotationally symmetric solutions to Ricci flow on $\mathbb{R}^{N}$ which encounter a global singularity at a finite time $T$. The singularity forms arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate $(T-t)^{-(\lambda+1)}$. Near the origin, blow-ups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blow-ups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on $\mathbb{R}^N$ whose blow-ups near the origin converge uniformly to the Bryant soliton.