Ben Andrews & Christopher Hopper 
Ricci Flow in Riemannian Geometry [PDF ebook] 
A Complete Proof of the Differentiable 1/4-Pinching Sphere Theorem

This book focuses on Hamilton’s Ricci flow, beginning with a detailed discussion of the required aspects of differential geometry, progressing through existence and regularity theory, compactness theorems for Riemannian manifolds, and Perelman’s noncollapsing results, and culminating in a detailed analysis of the evolution of curvature, where recent breakthroughs of Bohm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1/4-pinching sphere theorem.