Ricci flow of warped Berger metrics on $${\mathbb {R}}^{4}$$ R 4
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Calculus of Variations
Ricci flow of warped Berger metrics on R4 Francesco Di Giovanni1 Received: 1 November 2019 / Accepted: 31 July 2020 © The Author(s) 2020
Abstract We study the Ricci flow on R4 starting at an SU(2)-cohomogeneity 1 metric g0 whose restriction to any hypersphere is a Berger metric. We prove that if g0 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 of a non-rotationally symmetric Type-II flow converging to a rotationally symmetric singularity model. Next, we show that if instead g0 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. Mathematics Subject Classification 53E20
1 Introduction Given a smooth Riemannian manifold (M, g0 ), Hamilton’s Ricci flow starting at g0 is defined to be the geometric heat-type evolution equation [25] ∂g = −2Ricg(t) , g(0) = g0 . ∂t Shi proved that if (M, g0 ) is complete and has bounded curvature, then the Ricci flow problem admits a solution [41]. Moreover, such solution is unique in the class of complete solutions with bounded curvature by the work of Chen and Zhu [18]. A solution to the Ricci flow encounters a finite-time singularity at some T < ∞ if and only if [29,41] lim sup sup|Rmg(t) |g(t) = ∞. tT
M
Finite time singularities of the Ricci flow are classified as follows [29]: Type-I :
lim sup (T − t) sup |Rmg(t) |g(t) < ∞, tT
Type-II :
M
lim sup (T − t) sup |Rmg(t) |g(t) = ∞. tT
M
Communicated by P. Topping.
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Francesco Di Giovanni [email protected] Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 0123456789().: V,-vol
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F. D. Giovanni
According to results of Naber [36] and Enders-Müller-Topping [20] any parabolic dilation of a Type-I Ricci flow at a singular point converges to a non-flat gradient shrinking soliton. On the other hand, far less is known about Type-II singularities. The first examples of Type-II singularities in dimension n ≥ 3 were found in [24] by Gu and Zhu, who considered a family of rotationally invariant Ricci flows on S n . 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 Rn by Wu in [44] and later, for a larger set of initial data, by the author in [19]. Only very recently the first explicit examples of non rotationally symmetric Type-II Ricci flows in dimension higher than three have been analysed by Appleton in [4] and by Stolarski in [43], where they both obtained Ricci flat singularity models. In our first result we show that a large family of 4-dimensional cohomogeneity 1 Ricci flows develop Type-II singularities modelled on the
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