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
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