Four-dimensional cohomogeneity one Ricci flow and nonnegative sectional curvature

Bettiol, Renato G.; Krishnan, Anusha M.

doi:10.4310/cag.2019.v27.n3.a1

Cited by 11 publications

(18 citation statements)

References 34 publications

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“…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.…”

mentioning

confidence: 99%

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

Giovanni

2020

Calc. Var.

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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.

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confidence: 99%

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

Giovanni

2020

Calc. Var.

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“…In fact, we can find initial data with nonnegative sectional curvature flowing to the Ricci-flat Taub-NUT metric. While the fact that positive sectional curvature is not preserved along the flow in dimension higher than three is well known, even in the cohomogeneity-1 setting [22], in the result below we prove that negative sectional curvature terms not only appear along the solution but also balance out the positive terms to yield a Ricci-flat limit in infinite time.…”

Section: Introductionmentioning

confidence: 46%

“…(ii) ( 21) ) = (20): it suffices to consider the following example By the existence of the steady soliton found by Appleton we know that Ricci flow solutions starting at initial data as in Definition 3 with k ¼ 1 might in general fail to converge to the Taub-NUT metric. We also note that the Euclidean metric would be included in the class G 0 if we dropped the requirement on the size of the Hopf-fiber in (22).…”

Section: Initial Data Opening Faster Than a Paraboloidmentioning

confidence: 99%

Convergence of Ricci flow solutions to Taub-NUT

Giovanni

2021

Communications in Partial Differential Equations

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We study the Ricci flow starting at an SU(2) cohomogeneity-1 metric g 0 on R 4 with monotone warping coefficients and whose restriction to any hypersphere is a Berger metric. If g 0 has bounded Hopf-fiber, curvature controlled by the size of the orbits and opens faster than a paraboloid in the directions orthogonal to the Hopf-fiber, then the flow converges to the Taub-NUT metric g TNUT in the Cheeger-Gromov sense in infinite time. We also classify the long-time behaviour when g 0 is asymptotically flat. In order to identify infinite-time singularity models we obtain a uniqueness result for g TNUT : ARTICLE HISTORY

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“…Theorem A builds on our earlier result [BK19] that certain metrics with sec ≥ 0, introduced by Grove and Ziller [GZ00] in a much broader context (see Section 2.1.1), immediately acquire negatively curved planes on S 4 and CP 2 , when evolved under Ricci flow. In light of the appropriate continuous dependence of Ricci flow on its initial data [BGI20], the metrics in Theorem A are obtained by means of: Theorem B.…”

Section: Introductionmentioning

confidence: 70%