532
conjugate heat equation, 201
connected sum decomposition, 433
convergence
of maps
C^00 on compact sets, 155
CP, 155
convergence on JR.^2 , 478
convex, 178
convex function, 367
coupled modified Ricci flow, 198
cross curvature breather, 499
cross curvature flow, 492
monotonicity formula, 496
short-time existence, 493
cross curvature soliton, 499
cross curvature tensor, 491
88-Lemma, 70
derivation of a Lie algebra, 33
derivative
first, 366
second, 366
derivative of metric bounds, 132
diameter control, 268
differentiable function, 364
dilaton, 49, 191
dilaton shift, 49
dimension reduction, 440, 472
direct limit, 156
directed system, 156
Dirichlet boundary condition, 269
doubly-warped product, 26, 50effective action, 49
Einstein metric, 5, 53
homogeneous, 32
standard type, 32
Einstein summation convention, 59
extended, 67
energy
of a path, 286
energy functional, 44, 191
entropy, 124
entropy functional, 222
Euler-Lagrange equation for
minimizer, 237
existence of minimizer, 238
entropy monotonicity
for gradient flow, 226
for Ricci flow, 226
s-entropy, 229
essential 2-sphere, 433
eternal solution, L165INDEXEuclidean space
characterization of, 451
exhaustion, 128
expanding breathers
nonexistence, 213
expanding gradient soliton, 46
expanding soliton, 2
exploding soliton, 14
exponential map
derivatives of, 175
exterior differential systems, 10
F-functional, 191
Fano manifold, 72
first Chern class, 64
first derivative, 366
first fundamental form, 482
first variation
of F, 192
of W, 223
of We, 231
Frankel Conjecture, 73
fundamental solution
of the heat equation, 453
funny way, 282
Futaki functional, 73, 98
Futaki invariant, 72
Gaussian soliton, 5
geometrically atoroidal, 436
geometrization conjecture, 437
gradient flow of :F"', 197
gradient soliton structure, 4
graph
space-time, 292
graph manifold
Cheeger-Gromov, 439
topological, 438
Gromov-Hausdorff convergence, 38
Gross's logarithmic Sobolev inequality
Beckner-Pearson's proof, 249
Haken 3-manifold, 436
Hamilton's surface entropy
formula, 217
gradient is matrix Harnack, 218
monotonicity, 217
harmonic map heat flow, 490
monotonicity formula, 490
Harnack inequality
classical, 91
heat equation, 453
fundamental solution, 453
Hermitian metric, 57