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A Survey of Einstein Metrics on 4-manifolds&Michael T.Anderson1

1 Introduction1

2 Brief review:4-manifolds,complex surfaces and Einstein metrics2

3 Constructions of Einstein metrics Ⅰ5

4 Obstructions to Einstein metrics9

5 Moduli spaces Ⅰ13

6 Moduli spaces Ⅱ25

7 Constructions of Einstein metrics Ⅱ29

8 Concluding remarks35

References35

Sphere Theorems in Geometry&Simon Brendle,Richard Schoen41

1 The Topological Sphere Theorem41

2 Manifolds with positive isotropic curvature42

3 The Differentiable Sphere Theorem53

4 New invariant curvature conditions for the Ricci flow56

5 Rigidity results and the classification of weakly 1/4-pinched manifolds63

6 Hamilton's differential Harnack inequality for the Ricci flow67

7 Compactness of pointwise pinched manifolds68

References72

Curvature Flows and CMC Hypersurfaces&Claus Gerhardt77

1 Introduction77

2 Notations and preliminary results77

3 Evolution equations for some geometric quantities80

4 Essential parabolic flow equations85

5 Existence results91

6 Curvature flows in Riemannian manifolds104

7 Foliation of a spacetime by CMC hypersurfaces112

8 The inverse mean curvature flow in Lorentzian spaces123

References125

Geometric Structures on Riemannian Manifolds&Naichung Conan Leung129

1 Introduction129

2 Topology of manifolds131

2.1 Cohomology and geometry of differential forms131

2.2 Hodge theorem134

2.3 Witten-Morse theory137

2.4 Vector bundles and gauge theory138

3 Riemannian geometry143

3.1 Torsion and Levi-Civita connections143

3.2 Classification of Riemannian holonomy groups144

3.3 Riemannian curvature tensors145

3.4 Flat tori146

3.5 Einstein metrics149

3.6 Minimal submanifolds149

3.7 Harmonic maps151

4 Oriented four manifcllds152

4.1 Gauge theory in dimension four153

4.2 Riemannian geometry in dimension four155

5 K？hler geometry156

5.1 K？hler geometry—complex aspects157

5.2 K？hler geometry—Riemannian aspects161

5.3 K？hler geometry—symplectic aspects165

5.4 Gromov-Witten theory168

6 Calabi-Yau geometry170

6.1 Calabi-Yau manifolds170

6.2 Special Lagrangian geometry172

6.3 Mirror symmetry174

6.4 K3 surfaces180

7 Calabi-Yau 3-folds183

7.1 Moduli of CY threefolds183

7.2 Curves and surfaces in Calabi-Yau threefolds185

7.3 Donaldson-Thomas bundles over Calabi-Yau threefolds188

7.4 Special Lagrangian submanifolds in CY3189

7.5 Mirror symmetry for Calabi-Yau threefolds189

8 G2-geometry190

8.1 G2-manifolds190

8.2 Moduli of G2-manifolds192

8.3 (Co-)associative geometry193

8.4 G2-Donaldson-Thomas bundles195

8.5 G2-dualities,trialities and M-theory196

9 Geometry of vector cross products197

9.1 VCP manifolds197

9.2 Instantons and branes199

9.3 Symplectic geometry on higher dimensional knot spaces200

9.4 C-VCP geometry200

9.5 Hyperk？hler geometry on isotropic knot spaces of CY201

10 Geometry over normed division algebras203

10.1 Manifolds over normed algebras203

10.2 Gauge theory over(special)A-manifolds205

10.3 A-submanifolds and(special)Lagrangian submanifolds205

11 Quaternion geometry207

11.1 Hyperk？hler geometry208

11.2 Quaternionic-K？hler geometry212

12 Conformal geometry212

13 Geometry of Riemannian symmetric spaces215

13.1 Riemannian symmetric spaces215

13.2 Jordan algebras and magic square217

13.3 Hermitian and quaternionic symmetric spaces219

14 Conclusions221

References222

Symplectic Calabi-Yau Surfaces&Tian-Jun Li231

1 Introduction231

2 Linear symplectic geometry233

2.1 Symplectic vector spaces233

2.2 Compatible complex structures235

2.3 Hermitian vector spaces238

2.4 4-dimensional geometry241

3 Symplectic manifolds245

3.1 Almost symplectic and almost complex structures245

3.2 Cohomological invariants and space of symplectic structures247

3.3 Moser stability and Darboux charts251

3.4 Submanifolds and their neighborhoods253

3.5 Constructions254

4 Almost K？hler geometry259

4.1 Almost Hermitian manifolds,integrability and operators259

4.2 Levi-Civita connection263

4.3 Connections and curvature on Hermitian bundles266

4.4 Chern connection and Hermitian curvatures271

4.5 The self-dual operator275

4.6 Dirac operators276

4.7 Weitzenb？ck formulas and some almost K？hler identities281

5 Seiberg-Witten theory-three facets283

5.1 SW equations284

5.2 Pin(2)symmetry for a spin reduction289

5.3 The compactness and Hausdorff property of the moduli space295

5.4 Generic smoothness of the moduli space298

5.5 Furuta's finite dim.Approximations302

5.6 SW invariants311

5.7 Symplectic SW equations and Taubes'nonvanishing313

5.8 Symplectic SW solutions and Pseudo-holomorphic curves319

5.9 Bordism SW invariants via finite dim.Approximations321

5.10 Mod 2 vanishing and homology type327

6 Symplectic Calabi-Yau equation333

6.1 Uniqueness and openness334

6.2 A priori estimates335

7 Symplectic Calabi-Yau surfaces337

7.1 Symplectic birational geometry and Kodaira dimension337

7.2 Examples338

7.3 Homological classification344

7.4 Further questions348

References352

Lectures on Stability and Constant Scalar Curvature&D.H.Phong,Jacob Sturm357

1 Introduction357

2 The conjecture of Yau360

2.1 Constant scalar curvature metrics in a given K？hler class360

2.2 The special case of K？hler-Einstein metrics361

2.3 The conjecture of Yau361

3 The analytic problem362

3.1 Fourth order non-linear PDE and Monge-Ampère equations362

3.2 Geometric heat flows363

3.3 Variational formulation and energy functionals363

4 The spaces Kk of Bergman metrics365

4.1 Kodaira imbeddings365

4.2 The Tian-Yau-Zelditch theorem366

5 The functional F0ω0(φ)on Kk368

5.1 F0ω0 and balance imbeddings369

5.2 F0ω0 and the Euler-Lagrange equation R-？=0370

5.3 F0ω0 and Monge-Ampère masses371

6 Notions of stability372

6.1 Stability in GIT372

6.2 Donaldson's infinite-dimensional GIT381

6.3 Stability conditions on Diff(X)orbits383

7 The necessity of stability385

7.1 The Moser-Trudinger inequality and analytic K-stability385

7.2 Necessity of Chow-Mumford stability387

7.3 Necessity of semi K-stability391

8 Sufficient conditions:the K？hler-Einstein case394

8.1 The α-invariant395

8.2 Nadel's multiplier ideal sheaves criterion395

8.3 The K？hler-Ricci flow397

9 General L:energy functionals and Chow points408

9.1 F0ω and Chow points408

9.2 Kω and Chow points410

10 General L:the Calabi energy and the Calabi flow411

10.1 The Calabi flow411

10.2 Extremal metrics and stability412

11 General L:toric varieties414

11.1 Symplectic potentials415

11.2 K-stability on toric varieties415

11.3 The K-unstable case419

12 Geodesics in the space K of K？hler potentials419

12.1 The Dirichlet problem for the complex Monge-Ampère equation419

12.2 Method of elliptic regularization and a priori estimates420

12.3 Geodesics in K and geodesics in Kk423

References427

Analytic Aspect of Hamilton's Ricci Flow&Xi-Ping Zhu437

Introduction437

1 Short-time existence and uniqueness438

2 Curvature estimates441

2.1 Shi's local derivative estimates442

2.2 Preserving positive curvature443

2.3 Hamilton-Ivey pinching estimate444

2.4 Li-Yau-Hamilton inequality448

3 Singularities of solutions450

3.1 Can all types of singularities be formed450

3.2 Singularity models452

3.3 Canonical neighborhood structure456

4 Long time behaviors457

4.1 The Ricci flow on two-manifolds458

4.2 The Ricci flow on three-manifolds461

4.3 Differential Sphere Theorems464

References468