A Survey of Einstein Metrics on 4-manifolds&Michael T.Anderson 1
1 Introduction 1
2 Brief review:4-manifolds,complex surfaces and Einstein metrics 2
3 Constructions of Einstein metrics Ⅰ 5
4 Obstructions to Einstein metrics 9
5 Moduli spaces Ⅰ 13
6 Moduli spaces Ⅱ 25
7 Constructions of Einstein metrics Ⅱ 29
8 Concluding remarks 35
References 35
Sphere Theorems in Geometry&Simon Brendle,Richard Schoen 41
1 The Topological Sphere Theorem 41
2 Manifolds with positive isotropic curvature 42
3 The Differentiable Sphere Theorem 53
4 New invariant curvature conditions for the Ricci flow 56
5 Rigidity results and the classification of weakly 1/4-pinched manifolds 63
6 Hamilton's differential Harnack inequality for the Ricci flow 67
7 Compactness of pointwise pinched manifolds 68
References 72
Curvature Flows and CMC Hypersurfaces&Claus Gerhardt 77
1 Introduction 77
2 Notations and preliminary results 77
3 Evolution equations for some geometric quantities 80
4 Essential parabolic flow equations 85
5 Existence results 91
6 Curvature flows in Riemannian manifolds 104
7 Foliation of a spacetime by CMC hypersurfaces 112
8 The inverse mean curvature flow in Lorentzian spaces 123
References 125
Geometric Structures on Riemannian Manifolds&Naichung Conan Leung 129
1 Introduction 129
2 Topology of manifolds 131
2.1 Cohomology and geometry of differential forms 131
2.2 Hodge theorem 134
2.3 Witten-Morse theory 137
2.4 Vector bundles and gauge theory 138
3 Riemannian geometry 143
3.1 Torsion and Levi-Civita connections 143
3.2 Classification of Riemannian holonomy groups 144
3.3 Riemannian curvature tensors 145
3.4 Flat tori 146
3.5 Einstein metrics 149
3.6 Minimal submanifolds 149
3.7 Harmonic maps 151
4 Oriented four manifcllds 152
4.1 Gauge theory in dimension four 153
4.2 Riemannian geometry in dimension four 155
5 K？hler geometry 156
5.1 K？hler geometry—complex aspects 157
5.2 K？hler geometry—Riemannian aspects 161
5.3 K？hler geometry—symplectic aspects 165
5.4 Gromov-Witten theory 168
6 Calabi-Yau geometry 170
6.1 Calabi-Yau manifolds 170
6.2 Special Lagrangian geometry 172
6.3 Mirror symmetry 174
6.4 K3 surfaces 180
7 Calabi-Yau 3-folds 183
7.1 Moduli of CY threefolds 183
7.2 Curves and surfaces in Calabi-Yau threefolds 185
7.3 Donaldson-Thomas bundles over Calabi-Yau threefolds 188
7.4 Special Lagrangian submanifolds in CY3 189
7.5 Mirror symmetry for Calabi-Yau threefolds 189
8 G2-geometry 190
8.1 G2-manifolds 190
8.2 Moduli of G2-manifolds 192
8.3 (Co-)associative geometry 193
8.4 G2-Donaldson-Thomas bundles 195
8.5 G2-dualities,trialities and M-theory 196
9 Geometry of vector cross products 197
9.1 VCP manifolds 197
9.2 Instantons and branes 199
9.3 Symplectic geometry on higher dimensional knot spaces 200
9.4 C-VCP geometry 200
9.5 Hyperk？hler geometry on isotropic knot spaces of CY 201
10 Geometry over normed division algebras 203
10.1 Manifolds over normed algebras 203
10.2 Gauge theory over(special)A-manifolds 205
10.3 A-submanifolds and(special)Lagrangian submanifolds 205
11 Quaternion geometry 207
11.1 Hyperk？hler geometry 208
11.2 Quaternionic-K？hler geometry 212
12 Conformal geometry 212
13 Geometry of Riemannian symmetric spaces 215
13.1 Riemannian symmetric spaces 215
13.2 Jordan algebras and magic square 217
13.3 Hermitian and quaternionic symmetric spaces 219
14 Conclusions 221
References 222
Symplectic Calabi-Yau Surfaces&Tian-Jun Li 231
1 Introduction 231
2 Linear symplectic geometry 233
2.1 Symplectic vector spaces 233
2.2 Compatible complex structures 235
2.3 Hermitian vector spaces 238
2.4 4-dimensional geometry 241
3 Symplectic manifolds 245
3.1 Almost symplectic and almost complex structures 245
3.2 Cohomological invariants and space of symplectic structures 247
3.3 Moser stability and Darboux charts 251
3.4 Submanifolds and their neighborhoods 253
3.5 Constructions 254
4 Almost K？hler geometry 259
4.1 Almost Hermitian manifolds,integrability and operators 259
4.2 Levi-Civita connection 263
4.3 Connections and curvature on Hermitian bundles 266
4.4 Chern connection and Hermitian curvatures 271
4.5 The self-dual operator 275
4.6 Dirac operators 276
4.7 Weitzenb？ck formulas and some almost K？hler identities 281
5 Seiberg-Witten theory-three facets 283
5.1 SW equations 284
5.2 Pin(2)symmetry for a spin reduction 289
5.3 The compactness and Hausdorff property of the moduli space 295
5.4 Generic smoothness of the moduli space 298
5.5 Furuta's finite dim.Approximations 302
5.6 SW invariants 311
5.7 Symplectic SW equations and Taubes'nonvanishing 313
5.8 Symplectic SW solutions and Pseudo-holomorphic curves 319
5.9 Bordism SW invariants via finite dim.Approximations 321
5.10 Mod 2 vanishing and homology type 327
6 Symplectic Calabi-Yau equation 333
6.1 Uniqueness and openness 334
6.2 A priori estimates 335
7 Symplectic Calabi-Yau surfaces 337
7.1 Symplectic birational geometry and Kodaira dimension 337
7.2 Examples 338
7.3 Homological classification 344
7.4 Further questions 348
References 352
Lectures on Stability and Constant Scalar Curvature&D.H.Phong,Jacob Sturm 357
1 Introduction 357
2 The conjecture of Yau 360
2.1 Constant scalar curvature metrics in a given K？hler class 360
2.2 The special case of K？hler-Einstein metrics 361
2.3 The conjecture of Yau 361
3 The analytic problem 362
3.1 Fourth order non-linear PDE and Monge-Ampère equations 362
3.2 Geometric heat flows 363
3.3 Variational formulation and energy functionals 363
4 The spaces Kk of Bergman metrics 365
4.1 Kodaira imbeddings 365
4.2 The Tian-Yau-Zelditch theorem 366
5 The functional F0ω0(φ)on Kk 368
5.1 F0ω0 and balance imbeddings 369
5.2 F0ω0 and the Euler-Lagrange equation R-？=0 370
5.3 F0ω0 and Monge-Ampère masses 371
6 Notions of stability 372
6.1 Stability in GIT 372
6.2 Donaldson's infinite-dimensional GIT 381
6.3 Stability conditions on Diff(X)orbits 383
7 The necessity of stability 385
7.1 The Moser-Trudinger inequality and analytic K-stability 385
7.2 Necessity of Chow-Mumford stability 387
7.3 Necessity of semi K-stability 391
8 Sufficient conditions:the K？hler-Einstein case 394
8.1 The α-invariant 395
8.2 Nadel's multiplier ideal sheaves criterion 395
8.3 The K？hler-Ricci flow 397
9 General L:energy functionals and Chow points 408
9.1 F0ω and Chow points 408
9.2 Kω and Chow points 410
10 General L:the Calabi energy and the Calabi flow 411
10.1 The Calabi flow 411
10.2 Extremal metrics and stability 412
11 General L:toric varieties 414
11.1 Symplectic potentials 415
11.2 K-stability on toric varieties 415
11.3 The K-unstable case 419
12 Geodesics in the space K of K？hler potentials 419
12.1 The Dirichlet problem for the complex Monge-Ampère equation 419
12.2 Method of elliptic regularization and a priori estimates 420
12.3 Geodesics in K and geodesics in Kk 423
References 427
Analytic Aspect of Hamilton's Ricci Flow&Xi-Ping Zhu 437
Introduction 437
1 Short-time existence and uniqueness 438
2 Curvature estimates 441
2.1 Shi's local derivative estimates 442
2.2 Preserving positive curvature 443
2.3 Hamilton-Ivey pinching estimate 444
2.4 Li-Yau-Hamilton inequality 448
3 Singularities of solutions 450
3.1 Can all types of singularities be formed 450
3.2 Singularity models 452
3.3 Canonical neighborhood structure 456
4 Long time behaviors 457
4.1 The Ricci flow on two-manifolds 458
4.2 The Ricci flow on three-manifolds 461
4.3 Differential Sphere Theorems 464
References 468