of degree 1, 428
injectivity radius estimate
Cheeger-Gromov-Taylor, 81
integral curvefor Y' f /IV fl^2 , 444
interior tangent cone, 418
intrinsic metric, 391
induced, 392
strictly, 391
isometric, 390
embedding, 390
isometry, 390
isometry group
is preserved, 212
isoperimetric inequality
Euclidean, 159
Jensen's inequality, 29h:-collapsed
strongly, 112
h:-gap theorem, 117
h:-noncollapsed
at all scales, 80
below the scale p, 80
h:-solution, 81
2-dimensional , 104
Perelman's, 94
satisfying trace Harnack, 89
with Harnack, 104
King-Rosenau solution, 82
£-distance, 107
£-length, 107
..-concave, 410
..-invariant, 2
large curvature points, 168
length space, 391
complete, 391
quasi-, 390
length structure
induced by a metric, 392
Levi
parametrix method, 263
Li-Schoen argument, 315
Li-Yau inequality, 318
limit
backward, 114
line, 407
linear trace Harnack quadratic, 31
Lipschitz
constant, 414
map, 413
INDEXlocalization
of Li-Yau inequality, 324
locally Lipschitz
m;;i,p, 414515logarithmic Sobolev inequality, 4, 206
Euclidean, 16, 206Matrix
The, ix
matrix Harnack
estimate, 89
quadratic, 32
maximum angle function, 476
mean value inequality
parabolic, 305
mean value property
space-time Euclidean , 366
space-time Riemannian, 369
metric
intrinsic, 391
strictly intrinsic, 391
subspace, 388
metric cone
Euclidean, 399
metric space, 388
complete, 388
induced by a pseudo-metric space,
389
pseudo-, 389
quasi-, 389
Minakshisundaram-Pleijel
method, 263
minimizer
existence of, 23
nonuniqueness of, 22
mollifier, 95
monotonicity formula
for entropy, 3
localized, 201
Moser iteration, 307
μ-invariant, 2
as 'T --+ 0, 16
continuous dependence on g, 12
is negative for 'T small, 15
lower bound, 7
upper bound, 5nearest point
projection map, 416
necks
rough monotonicity in positively
curved manifolds, 68
no local collapsing, 80