Ricci flow neckpinches without rotational symmetry

Abstract

We study "warped Berger" solutions S 1 × S 3 , G(t) of Ricci flow: generalized warped products with the metric induced on each fiber {s}×SU(2) a left-invariant Berger metric. We prove that this structure is preserved by the flow, that these solutions develop finite-time neckpinch singularities, and that they asymptotically approach round product metrics in space-time neighborhoods of their singular sets, in precise senses. These are the first examples of Ricci flow solutions without rotational symmetry that become asymptotically rotationally symmetric locally as they develop local finite-time singularities.

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

“…4 we study warped Berger Ricci flows evolving from initial data either in G or in G ∞ . Similarly to [31] we show that the curvature is controlled by the size of the principal orbits and that the solution becomes rotationally symmetric around any singularity at some rate that breaks scale-invariance. An important ingredient, for the case of G ∞ , is also given by the application of the Pseudolocality formula in [17] by Chau, Tam and Yu.…”

“…2.1 of [4]). In analogy with [31] we refer to any (U(2)-invariant) metric on R 4 of the form (3) and satisfying c ≤ b as a (generalized) warped Berger metric.…”

“…In our first result we show that a large family of 4-dimensional cohomogeneity 1 Ricci flows develop Type-II singularities modelled on the Bryant soliton [10]. The Ricci flow on 4-dimensional cohomogeneity 1 manifolds has been recently studied on various topologies [4,6,31,32]. In [31] Isenberg, Knopf and Šešum showed that the Ricci flow starting at a family of generalized warped Berger metrics on S 1 × S 3 is Type-I and becomes rotationally symmetric around any singularity.…”

“…In analogy with [31] we finally assume that c ≤ b so that each non-degenerate fiber {s} × S 3 is a Berger sphere. We call such metric a warped Berger metric on R 4 .…”

Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

“…This allows us to explicitly compute the first variation of the sectional curvature of certain initially flat planes and determine it is negative, hence the manifold immediately acquires some negatively curved planes under the flow. Similar cohomogeneity one frameworks were previously employed by Böhm [6] and Dancer and Wang [12] to construct Einstein metrics and Ricci solitons, by Pulemotov [30] to study Ricci flow on manifolds with boundary, and implicitly in several other recent works including [3,19,23].…”