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Ricci flow of warped Berger metrics on R⁴

Di Giovanni, F; (2020) Ricci flow of warped Berger metrics on R⁴. Calculus of Variations and Partial Differential Equations , 59 , Article 162. 10.1007/s00526-020-01823-4. Green open access

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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.

Type: Article
Title: Ricci flow of warped Berger metrics on R⁴
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s00526-020-01823-4
Publisher version: https://doi.org/10.1007/s00526-020-01823-4
Language: English
Additional information: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10117208
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