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arXiv · 1210.4089

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

Haotian Wu
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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.

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Haotian Wu. 2012-10-15. On type-II singularities in Ricci flow on $\mathbb{R}^{N}$. https://arxiv.org/abs/1210.4089

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