In this book, we study the evolution of Riemannian metrics under the
Ricci flow. This evolution equation was introduced in a seminal paper by
R. Hamilton [44], following earlier work of Eells and Sampson [33] on the
harmonic map heat flow. Using the Ricci flow, Hamilton proved that ev-
ery compact three-manifold with positive Ricci curvature is diffeomorphic
to a spherical space form. The Ricci flow has since been used to resolve
longstanding open questions in Riemannian geometry and three-dimensional
topology. In this text, we focus on the convergence theory for the Ricci flow
in higher dimensions and its application to the Differentiable Sphere Theo-
rem. The results we describe have all appeared in research articles. How-
ever, we have made an effort to simplify various arguments and streamline
the exposition.
In Chapter 1, we give a survey of various sphere theorems in Riemannian
geometry (see also [22]). We first describe the Topological Sphere Theorem
of Berger and Klingenberg. We then discuss various generalizations of that
theorem, such as the Diameter Sphere Theorem of Grove and Shiohama [42]
and the Sphere Theorem of Micallef and Moore [60]. These results rely on
the variational theory for geodesics and harmonic maps, respectively. We
will discuss the main ideas involved in the proof; however, this material will
not be used in later chapters. Finally, we state the Differentiable Sphere
Theorem obtained by the author and R. Schoen [20].
In Chapter 2, we state the definition of the Ricci flow and describe
the short-time existence and uniqueness theory. We then study how the
Riemann curvature tensor changes when the metric evolves under the Ricci
flow. This evolution equation will be the basis for all the a priori estimates
established in later chapters.
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