On Type-II Singularities in Ricci Flow on ℝ^N

Abstract

In each dimension N ≥ 3 and for each real number λ ≥ 1, we construct a family of complete rotationally symmetric solutions to Ricci flow on 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) −(λ+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 R N whose blow-ups near the origin converge uniformly to the Bryant soliton.

Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

“…We note the striking differences of our results from those of Angenent and Velázquez [2], who have constructed a set of MCF solutions on compact hypersurfaces which develop Type-II singularities with discrete "quantized" blow up rates of the form (T − t) 1/m−1 for integer m ≥ 3. These differences mirror the differences found between Type-II singularities in Ricci flow on noncompact manifolds [24] and those on compact manifolds [4,5].…”

“…As seen in a number of the works which study the detailed asymptotics of the formation of degenerate neckpinches in MCF or Ricci flow [2,4,5,24], a key first step in such a study is to use matched asymptotic analysis to produce formal approximate solutions of the flow. Formal solutions of this sort serve both as templates to which the actual solutions are shown to approach asymptotically, and as guides for the construction of subsolutions and supersolutions.…”

“…On compact manifolds Σ, all the examples that have been found [4,5] have "quantized" blowup rates of sup x∈Σ | Rm(x, t)| ∼ (T − t) 2 k −2 for integers k ≥ 3 (here T is the time of the first singularity). By contrast, for noncompact manifolds Σ, the known examples [24] have a continuous spectrum of blowup rates: sup To motivate how we build mean curvature flows of noncompact hypersurfaces which exhibit Type-II curvature blow-up, we consider the following setup (see Figure 1): Suppose we have a graph over an n-ball that asymptotically approaches the cylinder S n × R. If we evolve both the graph and the cylinder via MCF, then both surfaces will shrink: the cylinder shrinks to a line in finite time, and the evolving graph will move to the right and remain asymptotic to the shrinking cylinder. It follows from the work of Sáez and Schnürer [23] that the evolving graph disappears to spatial infinity at the same time as the cylinder collapses.…”

“…Hence, once we have shown (using a comparison principle) that any solution starting from initial data between the barriers does stay between them for all time, and once we have determined that such initial data sets do exist, we can conclude that there are MCF solutions which exhibit the behavior found in a region near the tip in the approximate solutions. This method of proof based on matched asymptotic analysis has been used in a number of studies of Type-I and Type-II singularities which develop both in Ricci flow [5,24] and in MCF [2,6].…”

Ricci flow of warped Berger metrics on $${\mathbb {R}}^{4}$$

“…Angenent, Isenberg and Knopf later discovered Type-II spherically symmetric Ricci flows on S n that are modelled on degenerate neckpinches [1]. Type-II singularities were also derived for rotationally invariant Ricci flows on R n by Wu in [44] and later, for a larger set of initial data, by the author in [19].…”