Preface

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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