Index

adjoint heat

kernel, 171

operator, 3, 90

Aleksandrov space, 403

cut locus, 409

exponential map, 408

splitting theorem, 407

α-large

curvature point, 168, 184

α-small

curvature point, 184

angle, 404

comparison, 404

arc length

evolution, 270

asymptotic scalar curvature ratio, 85,

124

asymptotic shrinker

existence of, 113

asymptotic volume ratio, 85, 124

is time-independent, 87

of Type I ancient solutions, 124

backward

limit, 118

solution to Ricci flow, 117

Bonnet–Myers theorem, 407

boundary

of Aleksandrov space, 409

boundedly compact, 397

Busemann function, 458

associated to a point, 458

Busemann–Feller theorem, 417

changing distances estimate, 45

application of, 101

Cheeger–Gromov

convergence, 174, 191

limits and heat kernels, 193

Cheeger–Gromov–Taylor

injectivity radius estimate, 81

cigar soliton, 153

Claim 1

on point picking, 170, 186

Claim 2

on point picking, 170, 188

compact modulo scaling, 137

compactness theorem

Perelman’s, 137

complete

geodesically, 409

concatenation

of two paths, 390

concave

function, 425

λ-, 410

semi-, 410

cone

Euclidean metric, 399

Riemannian, 86

topological, 399

convex function, 415, 425

convex set, 419

locally, 419

convolution

space-time, 229

curvature bump, 57

cut locus, 409

cylinder

parabolic, 184

Davies’ upper bound

for heat kernel, 361

defining equation

for HN , 218

derivative

directional, 428

derivative estimates

Perelman’s scaled, 150

Shi’s local, 144

513