| Preface |
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ix | |
| What this book is about |
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ix | |
| Highlights of Part I |
|
xi | |
| Acknowledgments |
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xiii | |
| Contents of Part I of Volume Two |
|
xvii | |
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1 | (54) |
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General solitons and their canonical forms |
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2 | (4) |
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Differentiating the soliton equation --- local and global analysis |
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6 | (5) |
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Warped products and 2-dimensional solitons |
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11 | (6) |
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Constructing the Bryant steady soliton |
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17 | (9) |
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Rotationally symmetric expanding solitons |
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26 | (6) |
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Homogeneous expanding solitons |
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32 | (9) |
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When breathers and solitons are Einstein |
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41 | (3) |
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Perelman's energy and entropy in relation to Ricci solitons |
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44 | (2) |
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Buscher duality transformation of warped product solitons |
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46 | (4) |
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Summary of results and open problems on Ricci solitons |
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50 | (2) |
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52 | (3) |
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Kahler--Ricci Flow and Kahler--Ricci Solitons |
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55 | (72) |
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Introduction to Kahler manifolds |
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55 | (7) |
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Connection, curvature, and covariant differentiation |
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62 | (8) |
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Existence of Kahler--Einstein metrics |
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70 | (4) |
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Introduction to the Kahler--Ricci flow |
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74 | (6) |
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Existence and convergence of the Kahler--Ricci flow |
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80 | (15) |
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Survey of some results for the Kahler--Ricci flow |
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95 | (2) |
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Examples of Kahler--Ricci solitons |
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97 | (6) |
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Kahler--Ricci flow with nonnegative bisectional curvature |
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103 | (6) |
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Matrix differential Harnack estimate for the Kahler--Ricci flow |
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109 | (9) |
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Linear and interpolated differential Harnack estimates |
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118 | (6) |
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124 | (3) |
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The Compactness Theorem for Ricci Flow |
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127 | (22) |
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Introduction and statements of the compactness theorems |
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127 | (5) |
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Convergence at all times from convergence at one time |
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132 | (6) |
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Extensions of Hamilton's compactness theorem |
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138 | (4) |
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Applications of Hamilton's compactness theorem |
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142 | (6) |
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148 | (1) |
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Proof of the Compactness Theorem |
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149 | (40) |
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149 | (1) |
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Approximate isometries, compactness of maps, and direct limits |
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150 | (8) |
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Construction of good coverings by balls |
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158 | (7) |
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The limit manifold (Mn∞, g∞) |
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165 | (10) |
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Center of mass and nonlinear averages |
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175 | (12) |
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187 | (2) |
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Energy, Monotonicity, and Breathers |
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189 | (32) |
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Energy, its first variation, and the gradient flow |
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190 | (7) |
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Monotonicity of energy for the Ricci flow |
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197 | (6) |
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Steady and expanding breather solutions revisited |
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203 | (11) |
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Classical entropy and Perelman's energy |
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214 | (5) |
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219 | (2) |
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Entropy and No Local Collapsing |
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221 | (64) |
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The entropy functional W and its monotonicity |
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221 | (14) |
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235 | (7) |
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Shrinking breathers are shrinking gradient Ricci solitons |
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242 | (4) |
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Logarithmic Sobolev inequality |
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246 | (5) |
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No finite time local collapsing: A proof of Hamilton's little loop conjecture |
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251 | (13) |
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Improved version of no local collapsing and diameter control |
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264 | (9) |
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Some further calculations related to F and W |
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273 | (11) |
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284 | (1) |
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285 | (96) |
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The L-length and distance for a static metric |
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286 | (2) |
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The L-length and the L-distance |
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288 | (8) |
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The first variation of L-length and existence of L-geodesics |
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296 | (10) |
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The gradient and time-derivative of the L-distance function |
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306 | (6) |
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The second variation formula for L and the Hessian of L |
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312 | (10) |
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Equations and inequalities satisfied by L and l |
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322 | (13) |
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The l-function on Einstein solutions and Ricci solitons |
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335 | (10) |
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L-Jacobi fields and the L-exponential map |
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345 | (18) |
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Weak solution formulation |
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363 | (16) |
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379 | (2) |
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Applications of the Reduced Distance |
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381 | (52) |
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Reduced volume of a static metric |
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381 | (5) |
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Reduced volume for Ricci flow |
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386 | (13) |
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A weakened no local collapsing theorem via the monotonicity of the reduced volume |
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399 | (7) |
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Backward limit of ancient κ-solution is a shrinker |
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406 | (11) |
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Perelman's Riemannian formalism in potentially infinite dimensions |
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417 | (15) |
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432 | (1) |
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Basic Topology of 3-Manifolds |
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433 | (12) |
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Essential 2-spheres and irreducible 3-manifolds |
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433 | (2) |
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Incompressible surfaces and the geometrization conjecture |
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435 | (4) |
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Decomposition theorems and the Ricci flow |
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439 | (3) |
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442 | (3) |
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Appendix A. Basic Ricci Flow Theory |
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445 | (32) |
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445 | (11) |
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456 | (9) |
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Basic singularity theory for Ricci flow |
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465 | (5) |
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More Ricci flow theory and ancient solutions |
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470 | (4) |
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Classical singularity theory |
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474 | (3) |
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Appendix B. Other Aspects of Ricci Flow and Related Flows |
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477 | (24) |
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Convergence to Ricci solitons |
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477 | (5) |
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482 | (8) |
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490 | (10) |
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500 | (1) |
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501 | (12) |
| Bibliography |
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513 | (18) |
| Index |
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531 | |
Lisainfo e-raamatute kohta