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Chapter 1. Riemannian Geometry 1 §1 Introduction 1 §2. Metrics, connections, curvatures and covariant differentiation 2 §3. Basic formulas and identities in Riemannian geometry 10 §4. Exterior differential calculus and Bochner formulas 14 §5. Integration and Hodge theory 20 §6. Curvature decomposition and locally conformally flat manifolds 25 §7. Moving frames and the Gauss-Bonnet formula 32 §8. Variation of arc length, energy and area 41 §9. Geodesics and the exponential map 52 §10. Second fundamental forms of geodesic spheres 58 §11 Laplacian, volume and Hessian comparison theorems 67 §12. Proof of the comparison theorems 73 §13 Manifolds with nonnegative curvature 80 §14. Lie groups and left-invariant metrics 87 §15. Notes and commentary 89 Chapter 2. Fundamentals of the Ricci Flow Equation 95 5§ I Geometric flows and geometrization 96 §2. Ricci flow and the evolution of scalar curvature 98 §3. The maximum principle for heat-type equations 00 §4. The Einstein-Hilbert functional 104 §5. Evolution of geometric quantities 1.08 §6. DeTurck's trick and short time existence 113 §7. Reaction-diffusion equation for the curvature tensor 119 §8. Notes and commentary 123 Chapter 3. Closed 3-manifolds with Positive Ricci Curvature 127 1. Hamilton's 3-manifolds with positive Ricci curvature theorem 127 §2. The maximum principle for tensors 128 §3. Curvature pinching estimates 131 §4. Gradient bounds for the scalar curvature 136 S5. Curvature tends to constant 140 §6. Exponential convergence of the normalized flow 142 7. .Notes and commentary 149 Chapter 4. Ricci Solitons and Special Solutions 153 § 1 Gradient Ricci solitons 154 §2. Gaussian and cylinder solitons 157 §3. Cigar steady soliton 159 §4. Rosenau solution 162 §5. An expanding soliton 164 §6. Bryant soliton 167 §7. Homogeneous solutions 169 §8. The isometry group 175 §9 Notes and commentary 176 Chapter 5. Isoperimetric Estimates and No Local Collapsing 181 §1 Sobolev and logarithmic Sobolev inequalities 181 2. Evolution of the length of a geodesic 186 *3. Isoperimetric estimate for surfaces 188 §4. Perelman's no local collapsing theorem 190 §5. Geometric applications of no local collapsing 198 §6. 3-manifolds with positive Ricci curvature revisited 206 §7. Isoperimetric estimate for 3-dimensional Type I solutions 208 §8. Notes and commentary 211 Chapter 6. Preparation for Singularity Analysis 213 §1. Derivative estimates and long time existence 213 2. Proof of Shi's local first and second derivative estimates 218 3. Cheeger-Gromov-type compactness theorem for Ricci flow 233 §4. Long time existence of solutions with bounded Ricci curvature 237 §5. The Hamilton-Ivey curvature estimate 240 §6. Strong maximum principles and metric splitting 245 §7. Rigidity of 3-manifolds with nonnegative curvature 248 §8. Notes and commentary 250 Chapter 7. High-dimensional and Noncompact Ricci Flow 253 §1. Spherical space form theorem of Huisken-Margerin-Nishikawa 254 2. 4-manifolds with positive curvature operator 259 3. Ma.nifolds with nonnegative curvature operator 263 §4. The maximum principle on noncompact manifolds 272 §5. Complete solutions of the Ricci flow on noncompact manifolds 279 §6. Notes and commentary 286 Chapter 8. Singularity Analysis 291 §1. Singularity dilations and types 292 §2. Point picking and types of singularity models 297 §3. Geometric invariants of ancient solutions 307 §4. Dimension reduction 316 §5, Notes and commentary 326 Chapter 9. Ancient Solutions 327 §1. Classification of ancient solutions on surfaces 328 §2. Properties of ancient solutions that relate to their type 338 §3. Geometry at infinity of gradient Ricci solitons 353 §4. Injectivity radius of steady gradient Ricci solitons 364 §5. Towards a classification of 3-dimensional ancient solutions 368 §6. Classification of 3-dimensional shrinking Ricci solitons 375 §7. Summary and open problems 388 Chapter 10. Differential Harnack Estimates 391 §1. Harnack estimates for the heat and Laplace equations 392 §2. Harnack estimate on surfaces with y > 0 397 §,3. Linear trace and interpolated Harnack estimates on surfaces 401 S4. Hamilton's matrix Harnack estimate for the Ricci flow 405 §5. Proof of the matrix Harnack estimate 410 §6 Harnack and pinching estimates for linearized Ricci flow 415 §7. Notes and commentary 420 Chapter 11. Space-time Geometry 425 1. Space-time solution to the Ricei flow for degenerate metrics 426 §2. Space-time curvature is the matrix Harnack quadratic .33 3. P otentially infinite metrics and potentially infinite dimensions 434 §4. Renormalizing the space-time length yields the f-length 452 §5. Space-time DeTurck's trick and fixing the measure 453 <6. Notes and commentary 456

Library of Congress subject headings for this publication: Global differential geometry, Ricci flow, Riemannian manifolds