Hamilton's Ricci flow has attracted considerable attention since
its introduction in 1982, owing partly to its promise in addressing
the Poincare conjecture and Thurston's geometrization conjecture.
This book gives a concise introduction to the subject with the
hindsight of Perelman's breakthroughs from 2002/2003. After
describing the basic properties of, and intuition behind the Ricci
flow, core elements of the theory are discussed such as
consequences of various forms of maximum principle, issues related
to existence theory, and basic properties of singularities in the
flow. A detailed exposition of Perelman's entropy functionals is
combined with a description of Cheeger-Gromov-Hamilton compactness
of manifolds and flows to show how a 'tangent' flow can be
extracted from a singular Ricci flow. Finally, all these threads
are pulled together to give a modern proof of Hamilton's theorem
that a closed three-dimensional manifold which carries a metric of
positive Ricci curvature is a spherical space form.
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