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<https://discovery.ucl.ac.uk/id/eprint/10133514> <http://purl.org/dc/terms/title> "Type-II singularities and long-time convergence of rotationally symmetric Ricci flows"^^<http://www.w3.org/2001/XMLSchema#string> .
<https://discovery.ucl.ac.uk/id/eprint/10133514> <http://purl.org/ontology/bibo/abstract> "In this thesis we study singularity formation and long-time behaviour of families of cohomogeneity one Ricci flows.\r\nIn Chapter 1 we analyse the Ricci flow on R\r\nn+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\r\nminimal hyperspheres and is asymptotic to a cylinder. This proves a conjecture by Chow\r\nand Tian about Perelman’s standard solutions. Conversely, we show that if g0 has no\r\nminimal hyperspheres but its curvature decays at infinity, then the solution is immortal.\r\nIn Chapter 2 we study the Ricci flow on R\r\n4\r\nstarting at an SU(2)-cohomogeneity one\r\nmetric g0 whose restriction to any hypersphere is a Berger metric. We prove that if g0\r\nhas no necks and is bounded by a cylinder, then the solution develops a global Type-II\r\nsingularity and converges to the Bryant soliton when suitably dilated at the origin. This is\r\nthe first example in dimension n > 3 of a non-SO(n)-invariant Type-II flow converging to\r\nan SO(n)-invariant singularity model. We also give conditions for the flow to be immortal\r\nand prove that if the solution is Type-I and controlled at spatial infinity, then there exist\r\nminimal 3-spheres for times close to the maximal time.\r\nIn Chapter 3 we focus the analysis on the class of immortal Ricci flows derived in\r\nChapter 2. We prove that if the initial metric has bounded Hopf-fiber, curvature controlled\r\nby the size of the orbits and opens faster than a paraboloid in the directions orthogonal to\r\nthe Hopf-fiber, then the flow converges to the Taub-NUT metric in the Cheeger-Gromov\r\nsense in infinite time. We also obtain a uniqueness result for Taub-NUT in a class of\r\ncollapsed ancient solutions."^^<http://www.w3.org/2001/XMLSchema#string> .
<https://discovery.ucl.ac.uk/id/eprint/10133514> <http://purl.org/dc/terms/date> "2021-08-28" .
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