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

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