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

Abstract

In each dimension $N\ge 3$ and for each real number $\lambda \ge 1$, 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.

Revised version, typos corrected

References in corpus (3)

• Ricci flow with surgery on three-manifolds
• Rotational symmetry of Ricci solitons in higher dimensions
• Degenerate neckpinches in Ricci flow