Index almost flat manifold, 17 ancient solution, 28 ancient, 235 ancient Type I essential point, 272 asymptotic analysis of singularities, 40 avoidance set, 102, 270 BBS derivative estimates, 116, 201, 223 Berger collapsed sphere, 11 Bianchi identities, 72 Bochner-Weitzenbock formula, 144, 195, 300 Busemann function associated to a point, 305 associated to a ray, 304 cigar soliton, 24, 168 collapse, 11 Compactness Theorem, 168, 231, 240 component representation of tensors, 279 conformal change of metric, 106 conjugate point, 289 conjugate radius, 290 connection, 280 covariant derivative, 280 critical point, 289 critical value, 289 cross curvature tensor, 87 cross-curvature flow, 88 curvature blow-up rate blow-up rate, 234 lower bound for, 240 curvature controlled sequence locally, 238 curvature essential sequence locally, 238 curve shortening flow, 160 cut locus, 296 in the tangent space, 298 cut point along a geodesic, 296 degnerate neckpinch, 63, 237, 244 DeTurck trick, 79 differential Harnack estimates, 132, 143, 148, 274 differential Harnack quantity for curvature of variable sign, 146, 155, 169 for positive curvature, 144 dimension reduction, 237, 250, 255, 262 divergence, 69, 75, 282 formal adjoint of, 75, 283 doubling-time estimate, 138, 225 Einstein operator, 83 elliptic differential operator, 72 elliptic PDE local solvability of, 91 Elliptization Conjecture, 3 entropy, 277 for a surface with mixed curvature, 135 for a surface with positive curvature, 133 for positive curvature, 141 eternal solution, 24 eternal, 235 evolution of length, 70 exterior derivative, 282, 283 formal adjoint of, 283 fixed point of Ricci flow, 21 focal point, 293 Gauss Lemma, 287 Gauss-Bonnet Theorem, 162, 166 generalized distance function, 300 geodesic k-gon, 150, 290 proper, 312 geometric structure, 4 geometrically atoroidal manifold, 2 Geometrization Conjecture, 3 global injectivity radius estimate 323

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