The Ricci Flow in Riemannian Geometry : A Complete Proof of the Differentiable 1 4 Pinching Sphere Theorem 1 Introduction 2 Background Material 3 Harmonic Mappings 4 Evolution of the Curvature 5 Short Time Existence 6 Uhlenbeck’s Trick 7 The Weak Maximum Principle 8 Regularity and Long Time Existence 9 The Compactness Theorem for Riemannian Manifolds 10 The F Functional and Gradient Flows 11 The W Functional and Local Noncollapsing 12 An Algebraic Identity for Curvature Operators 13 The Cone Construction of Böhm and Wilking 14 Preserving Positive Isotropic Curvature 15 The Final Argument This book focuses on Hamiltons 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 Perelmans noncollapsing results and culminating in a detailed analysis of the evolution of curvature where recent breakthroughs of Böhm and Wilking and Brendle and Schoen have led to a proof of the differentiable 1 4 pinching sphere theorem