Title: Ricci flow on cone surfaces and a three-dimensional expanding soliton,
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.