534
for a Kahler metric, 70, 106, 118
Lie algebra cohomology, 33
Lie algebra square, 460
Lie derivative, 449
linear trace Harnack
and W.,, 233
linearized Kahler-Ricci flow, 118
Lipschitz graph, 365
Little Loop Conjecture, 256, 257
locally bounded variation, 367
locally collapsing, 255
locally homogeneous geometries, 438
locally Lipschitz, 364
vector field, 365
logarithmic Sobolev inequality, 246
of L. Gross, 247
mapping torus, 37, 38
matrix Harnack formula
for adjoint heat equation, 281
matrix Harnack quadratic, 9
for Kahler-Ricci flow, 109
maximal function, 265
maximum principle
strong tensor, 107, 471
weak scalar, 458
weak tensor, 459
maximum volume growth, 96, 253, 507
mean curvature, 482
mean curvature flow, 482
convergence to self-similar, 489
evolution equations under, 483
measure, 196
minimal £-geodesic, 293
existence, 305
minimizer of :F
Euler-Lagrange equation for, 206
existence of, 205
modified Ricci curvature, 194
modified Ricci flow, 197
modified Ricci tensor, 48
modified scalar curvature, 48, 194
evolution of, 203
first variation, 273
modified subsequence convergence, 477
mollifier, 367
Monge-Ampere equation
complex, 75, 76
Monge-Ampere equation
complex, 71
monotonicity
of W.,, 233
monotonicity formula
INDEXfor static metric reduced volume , 382
for the gradient flow, 197
μ-invariant, 236
as r-+ O, 244
as r -+ oo, 243
finiteness, 237
monotonicity, 239
under Cheeger-Gromov convergence,
240necklike point, 470
neckpinch, 439
Newlander-Nirenberg Theorem, 56
Nijenhuis tensor, 56
nilpotent Lie group, 34
NKRF, 81
no local collapsing, 256, 440
improved version, 267
proof of, 258
normal holomorphic coordinates, 68
normal surfaces, 435
v-invariant, 236
monotonicity, 244
null eigenvector assumption, 459open problem, 282
orbifold, 12, 141
bad, 12parabola, 287
parallel 2-spheres, 434
Perelman's energy functional, 44
Perelman's entropy functional, 45, 222
Perelman's equations
coupled to Ricci flow, 225
Perelman's Harnack quantity, 227
evolution, 227
pinching improves, 463
PL category, 433
Poincare conjecture, 437
pointed Riemannian manifold, 129
pointed solution to the Ricci flow, 129
pointwise convergence
in GP, 128
uniformly on compact sets, 129
pointwise monotonicity
along £-geodesics, 391
porous medium equation, 479
potential function, 76
potentially infinite dimensions
and conformal geometry, 194quasi-Einstein metric, 52