eprintid: 10133514
rev_number: 28
eprint_status: archive
userid: 608
dir: disk0/10/13/35/14
datestamp: 2021-10-07 11:40:35
lastmod: 2021-10-08 21:40:48
status_changed: 2021-10-07 11:40:35
type: thesis
metadata_visibility: show
creators_name: Di Giovanni, Francesco
title: Type-II singularities and long-time convergence of rotationally symmetric Ricci flows
ispublished: unpub
divisions: UCL
divisions: B04
divisions: C06
note: 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.
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.
date: 2021-08-28
date_type: published
oa_status: green
full_text_type: other
thesis_class: doctoral_open
thesis_award: Ph.D
language: eng
thesis_view: UCL_Thesis
primo: open
primo_central: open_green
verified: verified_manual
elements_id: 1883685
lyricists_name: Di Giovanni, Francesco
lyricists_id: FDIGI43
actors_name: Di Giovanni, Francesco
actors_name: Allington-Smith, Dominic
actors_id: FDIGI43
actors_id: DAALL44
actors_role: owner
actors_role: impersonator
full_text_status: public
pages: 186
event_title: UCL (University College London)
institution: UCL (University College London)
department: Mathematics
thesis_type: Doctoral
citation:        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   
 
document_url: https://discovery.ucl.ac.uk/id/eprint/10133514/8/tesi_di_giovanni_.pdf