adjoint heat equation, 201, 228
Alexander's Theorem, 433
almost complex manifold, 56
almost complex structure, 56
integrable, 56
ancient solution, 465
anti-holomorphic tangent bundle, 58
aperture, 477
approximate isometry, 150
arc length
evolution, 457
variation formula, 448
Arzela-Ascoli Theorem, 137
associated 2-form, 61
asymptotic scalar curvature ratio, 472
is infinite on steady solitons, 4 72
asymptotic volume ratio, 253, 386, 451
of expanding soliton, 4 73
of Type I ancient solutions, 474
average
nonlinear, 182
average scalar curvature, 6backward heat equation, 198
backward Ricci flow, 288
Bakry-Emery log Sobolev inequality,
218
Bianchi identity
second, 66
bisectional curvature, 65
nonnegative, 95, 96, 103, 118
Bishop volume comparison theorem,
450
Bishop-Gromov volume comparison,
450
boundary condition
Dirichlet, 269
bounded geometry, 130
bounded variation
locally, 367
Index
531breather solution, 41
Bryant soliton, 17, 440
Buscher duality transformation, 46Calabi conjecture, 71
Calabi's trick, 453
Calabi-Yau metric, 72
canonical case, 73, 76
Cartan-Kahler Theorem, 10
center of mass, 182
Cheeger-Gromov convergence, 129
Cheeger-Gromov-Taylor, 159
Cheeger-Yau estimate, 453
Cheng's eigenvalue comparison, 269
Chevalley complex, 33
cigar soliton, 14, 102, 440, 468, 469
cigar-paraboloid behavior, 102
circumference at infinity, 4 77
Classic Dimension, xiv
classical entropy, 124, 214
monotonicity, 215
Cohn-Vossen inequality, 17
collapsible manifold, 442
compactness
of isometries, 155
compactness theorem
for Kahler metrics, 139
for Kahler-Ricci flow, 140
for metrics, 130
for solutions, 131, 138
local version, 138
complex manifold, 56
complex projective space, 98
complex structure, 56
complex submanifold, 56
complexified cotangent bundle, 58
complexified tangent bundle, 58
concatenated path, 292
conjugate
complex, 58