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

Abstract

AbstractWe study the Ricci flow on $${\mathbb {R}}^{4}$$ R 4 starting at an SU(2)-cohomogeneity 1 metric $$g_{0}$$ g 0 whose restriction to any hypersphere is a Berger metric. We prove that if $$g_{0}$$ g 0 has no necks and is bounded by a cylinder, then the solution develops a global Type-II singularity and converges to the Bryant soliton when suitably dilated at the origin. This is the first example in dimension $$n > 3$$ n > 3 of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead $$g_{0}$$ g 0 has no necks, its curvature decays and the Hopf fibres are not collapsed, then the solution is immortal. Finally, we prove that if the flow is Type-I, then there exist minimal 3-spheres for times close to the maximal time.