Index

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, 472

asymptotic volume ratio, 253, 386, 451

of expanding soliton, 473

of Type I ancient solutions, 474

average

nonlinear, 182

average scalar curvature, 6

backward 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

breather solution, 41

Bryant soliton, 17, 440

Buscher duality transformation, 46

Calabi 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, 477

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