514difference quotient, 428
differential Harnack estimate, 318
dilatation, 414
dimension
classic, xi
reduces, 53
reduction, 54
direction
of a path at a point, 405
directional derivative, 428
directions
space of, 408
Dirichlet heat kernel, 290
distance function
mollified, 488
time derivative of, 41
under Ricci flow, 42
distance-like function, 378
distance-preserving, 390
distortion
of a map, 394
domain
regular, 158
Duhamel's principle, 335energy functional
minimizer, 2
Perelman's, 2
entropy
existence of a minimizer, 23
maximum value of a minimizer, 31
minimizer may not be unique, 22
monotonicity formula, 3
monotonicity formula, localized, 201
entropy functional
Perelman's, 2, 90
c-isometry, 394
c--neck, 68
embedded, 62
in Bryant soliton, 83
c-neighborhood, 349, 393
c-net, 394
c-pointed Hausdorff approximation, 395
Euclidean isoperimetric inequality, 159
exponential map, 408
extremal set, 411forward difference quotient
lim inf of, 41
function
semi-concave, 410
fundamental solution
heat equation, 215
INDEXminimal positive, 216
fundamental theorem of calculus, 414generalized gradient, 433
geodesic, 391
geodesically complete, 409
gradient
generalized, 433
Gromoll-Meyer theorem, 63
Gromov-Hausdorff
convergence, 395
distance, 393
pointed convergence, 396
pointed distance, 395half-spaces, 459
Harnack estimate, 318
Perelman's, 201
Harnack quadratic
linear trace, 31
matrix, 32
trace, 32
Hausdorff
dimension, 405
distance, 393
measure, 405
heat ball, 365, 368
heat equation
fundamental solution, 215
heat kernel, 216, 334
asymptotics, 251
characterization of Ricci flow, 286
Davies' upper bound, 361
Dirichlet, 290
existence, 216
expansion, 232
first approximation to, 217
for time-dependent metric, 271
good approximation to, 217
£^1 -norm is preserved, 342
lower bound, 361
on noncompact manifold, 301
transplanted, 217
under Cheeger-Gromov limits, 193
upper bound, 356
heat operator
adjoint, 3, 266
heat sphere, 367, 373
Hessian
generalized, 456
hole
fixing, 387
homogenous