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