TY  - UNPB
PB  - UCL (University College London)
A1  - Di Giovanni, Francesco
N1  - 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.
UR  - https://discovery.ucl.ac.uk/id/eprint/10133514/
ID  - discovery10133514
EP  - 186
N2  - 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.
AV  - public
TI  - Type-II singularities and long-time convergence of rotationally symmetric Ricci flows
Y1  - 2021/08/28/
M1  - Doctoral
ER  -