Hessian upper bound, 370
holomorphic coordinates, 55
holomorphic sectional curvature, 65
holomorphic tangent bundle, 58
homogeneous space, 32
Huisken's monotonicity formula, 487
Hamilton's generalization, 489
hyperbolic space, 269immortal solution, 465
incompressible surface, 436
index form
Riemannian, 345
Index Lemma
Riemannian, 346
infinitesimal automorphism, 57, 97
injectivity radius estimate, 142, 159
local, 257
integral curves
of Vf, 329
integration by parts
for Lipschitz functions, 365
involutive system, 10
irreducible 3-manifold, 434
Jacobi field
Riemannian, 346
Jacobian
Riemannian, 359
Jense.n's inequality, 235
Kahler class, 61
Kahler form, 61
Kahler identities, 61
Kahler manifold, 57
Kahler metric, 57
Kahler-Einstein metric, 72
Kahler-Ricci fl.ow
normalized, 76
11;-collapsed
at the scale r, 252
11;-noncollapsed
below the scale p, 252
Killing vector field, 11
Kneser Finiteness Theorem, 434
Koiso soliton, 44, 98
KRF, 81
.C-cut locus, 356
L-distance, 293
and trace Harnack, 323
for Ricci fl.at solutions, 294
gradient of, 311
Hessian of, 318INDEX 533is locally. Lipschitz, 308
Laplacian Comparison Theorem, 321
time-derivative of, 312
triangle inequality, 328
£-exponential map, 352
as T --+ 0, 353
for Ricci fl.at solution, 353
£-geodesic, 298
estimate for speed of , 303
existence for IVP, 305
on Einstein solution, 299
£-geodesic equation, 298
£-geodesics
short ones are minimizing, 308
.C-Hopf-Rinow Theorem, 355
£-index form, 359
£-index lemma, 357
£-Jacobi equation, 347
£-Jacobi field, 346
time-derivative of, 351
£-Jacobian, 360
of Ricci fl.at solution, 361
time-derivative of, 362
£-length, 291
additivity, 292
first variation formula, 297
lower bound, 292
scaling property, 293
second variation of, 316
>.-invariant, 204
lower bound, 206
monotonicity, 209
second variation of, 280
upper bound, 206
Laplacian
for a Kahler metric, 67
Hodge-de Rham, 449
rough, 449
Laplacian comparison
Riemannian, 376
Laplacian Comparison Theorem, 452
for L-distance, 321
L-distance
as 7 --+ 0, 324
monotonicity of minimum, 324
supersolution to heat equation, 323
Levi-Civita connection, 446
evolution, 456
Kahler evolution, 77
of a Kahler metric, 62
variation formula, 448
Li-Yau inequality, 453
Lichnerowicz Laplacian, 448