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Type-II singularities and long-time convergence of rotationally symmetric Ricci flows

Di Giovanni, Francesco; (2021) Type-II singularities and long-time convergence of rotationally symmetric Ricci flows. Doctoral thesis (Ph.D), UCL (University College London). Green open access

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Abstract

In this thesis we study singularity formation and long-time behaviour of families of cohomogeneity one Ricci flows. In Chapter 1 we analyse the Ricci flow on R n+1, with n ≥ 2, starting at some complete bounded curvature SO(n + 1)-invariant metric g0. We prove that the solution develops a Type-II singularity and converges to the Bryant soliton after scaling if g0 has no minimal hyperspheres and is asymptotic to a cylinder. This proves a conjecture by Chow and Tian about Perelman’s standard solutions. Conversely, we show that if g0 has no minimal hyperspheres but its curvature decays at infinity, then the solution is immortal. In Chapter 2 we study the Ricci flow on R 4 starting at an SU(2)-cohomogeneity one 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-SO(n)-invariant Type-II flow converging to an SO(n)-invariant singularity model. We also give conditions for the flow to be immortal and prove that if the solution is Type-I and controlled at spatial infinity, then there exist minimal 3-spheres for times close to the maximal time. In Chapter 3 we focus the analysis on the class of immortal Ricci flows derived in Chapter 2. We prove that if the initial metric 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 in the Cheeger-Gromov sense in infinite time. We also obtain a uniqueness result for Taub-NUT in a class of collapsed ancient solutions.

Type: Thesis (Doctoral)
Qualification: Ph.D
Title: Type-II singularities and long-time convergence of rotationally symmetric Ricci flows
Event: UCL (University College London)
Open access status: An open access version is available from UCL Discovery
Language: English
Additional information: Copyright © The Author 2021. Original content in this thesis is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International (CC BY-NC 4.0) Licence (https://creativecommons.org/licenses/by-nc/4.0/). Any third-party copyright material present remains the property of its respective owner(s) and is licensed under its existing terms. Access may initially be restricted at the author’s request.
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
URI: https://discovery.ucl.ac.uk/id/eprint/10133514
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