Ricci flow on cone surfaces and a three-dimensional expanding soliton

TL;DR

This thesis advances the understanding of Ricci flow on cone surfaces by classifying solitons, extending geometrization to singular surfaces, demonstrating singularity removal, and constructing a unique 3D expanding soliton.

Contribution

It provides the first complete classification of Ricci solitons on cone surfaces, extends Ricci flow theory to singular cases, and constructs a unique 3D expanding soliton with specific properties.

Findings

Complete classification of 2D Ricci solitons on smooth and cone surfaces

Extension of Ricci flow to geometrize cone surfaces with singularities

Existence of Ricci flow solutions that remove cone singularities

Abstract

The main objective of this thesis is the study of the evolution under the Ricci flow of surfaces with singularities of cone type. A second objective, emerged from the techniques we use, is the study of families of Ricci flow solitons in dimension 2 and 3. The Ricci flow is an evolution equation for Riemannian manifolds, introduced by R. Hamilton in 1982. It is from the achievements made by G. Perelman with this technique in 2002 when the Ricci flow has been established in a discipline itself, generating a great interest in the community. This thesis contains four original results. First result is a complete classification of solitons in smooth and cone surfaces. This classification completes the preceding results found by Hamilton, Chow and Wu and others, and we obtain explicit descriptions of all solitons in dimension 2. Second result is a Geometrization of cone surfaces by Ricci flow. This result, which uses the aforementioned first result, extends the theory of Hamilton to the singular case. This is the most comprehensive result in the thesis, for which we use and develop analysis and PDE techniques, as well as comparison geometry techniques. Third result is the existence of a Ricci flow that removes cone singularities. This clearly exposes the non-uniqueness of solutions to the flow , in analogy to the Ricci flow with cusps of P. Topping. The fourth result is the construction of a new expanding gradient Ricci soliton in dimension 3. Just as we do with solitons on cone surfaces, we give an explicit construction using techniques of phase portraits. We also prove that this is the only soliton with its topology and its lower bound of the curvature, and besides this is a critical case amongst all expanding solitons in dimension 3 with curvature bounded below.