HAMILTON'S RICCI FLOW
CONTENTS CHAPTER 1 BASIC RIEMANNIAN GEOMETRY 1 INTRODUCTION 2 BASIC CONVENTIONS AND FORMULAS IN RIEMANNIAN GEOMETRY 3 LAPLACIAN AND HESSIAN COMPARISON THEOREMS 4 GEODESIC POLAR COORDINATES 5 FIRST AND SECOND VARIATION OF ARC LENGTH AND ENERGY FORMULAS 6 GEOMETRIC APPLICATIONS OF SECOND VARIATION 7 GREEN’S FUNCTION 8 COMPARISON THEORY FOR THE HEAT KERNEL 9 PARAMETRIX FOR THE HEAT EQUATION 10 EIGENVALUES AND EIGENFUNCTIONS OF THE LAPLACIAN 11 THE DETERMINANT OF THE LAPLACIAN 12 MONOTONICITY FOR HARMONIC FUNCTIONS AND MAPS 13 LIE GROUPS AND LEFT INVARIANT METRICS 14 BIEBERBACH THEOREM 15 COMPENDIUM OF INEQUALITIES 16 NOTES AND COMMENTARY CHAPTER 2 ELEMENTARY ASPECTS OF THE RICCI ?OW EQUATION 1 SOME GEOMETRIC ?OWS PREDATING RICCI ?OW 2 RICCI ?OW AND GEOMETRIZATION: A SHORT PREVIEW 3 RICCI ?OW AND THE EVOLUTION OF SCALAR CURVATURE 4 THE MAXIMUM PRINCIPLE FOR HEAT-TYPE EQUATIONS 5 THE MAXIMUM PRINCIPLE ON NONCOMPACT MANIFOLDS 6 THE EINSTEIN-HILBERT FUNCTIONAL 7 EVOLUTION OF GEOMETRIC QUANTITIES - LOCAL COORDINATE CALCULATIONS 8 DETURCK’S TRICK AND SHORT TIME EXISTENCE 9 NOTES AND COMMENTARY CONTENTS CHAPTER 3 CLOSED 3-MANIFOLDS WITH POSITIVE RICCI CURVATURE 1 THE MAXIMUM PRINCIPLE FOR TENSORS 2 HAMILTON’S 1982 THEOREM 3 EVOLUTION OF CURVATURE 4 THE MAXIMUM PRINCIPLE FOR SYSTEMS 5 GRADIENT OF SCALAR CURVATURE ESTIMATE 6 CURVATURE TENDS TO CONSTANT 7 EXPONENTIAL CONVERGENCE TO CONSTANT CURVATURE OF THE NORMALIZED ?OW 8 NOTES AND COMMENTARY CHAPTER 4 RICCI SOLITONS AND OTHER SPECIAL SOLUTIONS 1 TYPES OF LONG EXISTING SOLUTIONS 2 GRADIENT RICCI SOLITONS 3 GAUSSIAN SOLITON 4 CYLINDER SHRINKING SOLITON 5 CIGAR STEADY SOLITON 6 ROSENAU SOLUTION 7 AN EXPANDING SOLITON 8 BRYANT SOLITON 9 GEOMETRY AT SPATIAL IN?NITY OF ANCIENT SOLUTIONS 10 HOMOGENEOUS SOLUTIONS 11 ISOMETRY GROUP 12 NOTES AND COMMENTARY CHAPTER 5 ANALYTIC RESULTS AND TECHNIQUES 1 DERIVATIVE ESTIMATES AND LONG TIME EXISTENCE 2 CHEEGER-GROMOV TYPE COMPACTNESS THEOREM FOR RICCI ?OW 3 THE HAMILTON-IVEY CURVATURE ESTIMATE 4 STRONG MAXIMUM PRINCIPLES AND SPLITTING THEOREMS 5 3-MANIFOLDS WITH NONNEGATIVE CURVATURE 6 MANIFOLDS WITH NONNEGATIVE CURVATURE OPERATOR 7 NOTES AND COMMENTARY CHAPTER 6 SOME MISCELLANEOUS TECHNIQUES FOR THE RICCI, YAMABE AND CROSS CURVATURE ?OWS 1 KAZDAN-WARNER TYPE IDENTITIES AND YAMABE AND RICCI SOLITONS 2 ANDREWS’ POINCAR´E TYPE INEQUALITY 3 THE GRADIENT OF HAMILTON’S ENTROPY IS THE MATRIX HARNACK 4 THE YAMABE ?OW AND ALEKSANDROV RE?ECTION 5 ISOPERIMETRIC ESTIMATE 6 CROSS CURVATURE ?OW CHAPTER 7 INTRODUCTION TO SINGULARITIES 1 DILATING ABOUT A SINGULARITY AND TAKING LIMITS CONTENTS 2 SINGULARITY TYPES 3 DEGENERATE NECK PINCH 4 CLASSI?CATION OF ANCIENT SOLUTIONS ON SURFACES 5 EXTENDING NONCOMPACT ANCIENT SURFACE SOLUTIONS TO ETERNAL SOLUTIONS 6 DIMENSION REDUCTION 7 HAMILTON’S PARTIAL CLASSI?CATION OF 3-DIMENSIONAL SINGULARITIES 8 SOME CONJECTURES ABOUT ANCIENT SOLUTIONS CHAPTER 8 HARNACK TYPE ESTIMATES 1 LI-YAU ESTIMATE FOR THE HEAT EQUATION 2 SURFACES WITH POSITIVE CURVATURE 3 HARNACK ESTIMATE ON COMPLETE SURFACES WITH POSITIVE CURVATURE 4 LINEAR TRACE AND INTERPOLATED ESTIMATES FOR THE RICCI ?OW ON SURFACES 5 HAMILTON’S MATRIX ESTIMATE 6 SKETCH OF THE PROOF OF THE MATRIX HARNACK ESTIMATE 7 LINEAR TRACE HARNACK ESTIMATE IN ALL DIMENSIONS 8 A PINCHING ESTIMATE FOR SOLUTIONS OF THE LINEARIZED RICCI ?OW EQUATION CHAPTER 9 SPACE-TIME GEOMETRY 1 A SPACE-TIME SOLUTION TO THE RICCI ?OW FOR DEGENERATE METRICS 2 SPACE-TIME CURVATURE IS THE MATRIX HARNACK QUADRATIC 3 POTENTIALLY IN?NITE METRICS AND POTENTIALLY IN?NITE DIMENSIONS 4 RENORMALIZING THE SPACE-TIME METRIC YIELDS THE ?-LENGTH 5 SPACE-TIME DETURCK’S TRICK AND ?XING THE MEASURE 6 NOTES AND COMMENTARY