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

Abstract

We 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.

Convergence of Ricci flow solutions to Taub-NUT

“…In particular, we recap a few key properties of g TNUT : In Section 3 we focus on Ricci flows starting in G AF : In the asymptotically flat setting one can control the solution at spatial infinity in a precise way and hence maximum principle arguments follow. Similarly to other cohomogeneity-1 scenarios [6,[14][15][16], we prove that the curvature is uniformly controlled whenever the principal orbits are non-degenerate. More importantly, we show that if the Hopf-fiber is bounded, then the solution always opens faster than a paraboloid in R 3 in any space-time region where the roundness ratio c=b gets small.…”

“…The monotonicity condition is meant to generalise the lack of minimal embedded spheres for the SO(n)-invariant setting and is hence natural when the emphasis is on investigating the long-time behaviour of the Ricci flow. Indeed, in [16] we proved that the maximal complete, bounded curvature Ricci flow solution starting at some warped Berger metric with monotone coefficients and curvature decaying at spatial infinity is immortal. In fact, the result holds with assumptions weaker than the spatial monotonicity of both the coefficients b and c. However, the stronger requirement provided in Definition 2.2 allows us to control the injectivity radius of the solution only in terms of upper bounds of the curvature.…”

“…It is worth noting that the smoothness conditions reflect the underlying topology and hence lead to significant variations, both in terms of results and approach, when comparing the study of SU(2)U( 1)-invariant Ricci flows on different manifolds [14][15][16].…”

“…In [16] we proved that if g 0 is a complete warped Berger metric with monotone coefficients and curvature decaying at spatial infinity, then the maximal Ricci flow solution starting at g 0 is immortal. In light of such result and being g TNUT an asymptotically flat metric, we first focus on the following family of initial data: Definition 2.…”

“…We note that the notion of warped Berger metrics is consistent with the notations adopted in [14]. The Ricci flow problem in this symmetry class has been studied on different topologies and examples of non-rotationally symmetric Type-I and Type-II(a) singularities have been constructed in [14] and in [15,16] respectively. A well-known warped Berger metric on R 4 is given by the Taub-NUT metric g TNUT , which can be written explicitly as…”