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Volume Ⅰ 1

Commentary on 1

Métriques de K？hler-Einstein Sur les Variétés Ouvertes 1

Métriques de K？hler-Einstein Sur les Variétés Ouvertes&Shing-Tung Yau 3

Commentary on 7

The Classical Plateau Problem and the Topology of 3-manifolds 7

The Classical Plateau Problem and the Topology of 3-manifolds&William H.Meeks Ⅲ,Shing-Tung Yau 9

Commentary on 11

Geometric Bounds on the Low Eigenvalues of a Compact Surface 11

Geometric Bounds on the Low Eigenvalues of a Compact Surface&R.Schoen,S.Wolpert,and S.T Yau 13

Commentary on 21

Estimates of Eigenvalues of a Compact Riemannian Manifold 21

Estimates of Eigenvalues of a Compact Riemannian Manifold&Peter Li and Shing-Tung Yau 23

Commentary on 51

The Total Mass and the Topology of an Asymptotically Flat Space-time 51

The Total Mass and the Topology of an Asymptotically Flat Space-time&Shing-Tung Yau 55

Commentary on 61

The Role of Partial Differential Equations in Differential Geometry 61

The Role of Partial Differential Equations in Differential Geometry&Shing-Tung Yau 69

Commentary on 83

The Real Monge-Ampère Equation and Affine Flat Structures 83

The Real Monge-Ampère Equation and Affine Flat Structures&Shiu-Yuen Cheng and Shing-Tung Yau 85

Commentary on 103

Survey on Partial Differential Equations in Differential Geometry 103

Survey on Partial Differential Equations in Differential Geometry&Shing-Tung Yau 111

Commentary on 159

Problem Section 159

Problem Section(With Commentary)&Shing-Tung Yau 161

Commentary on 225

A Survey on K？hler-Einstein Metrics 225

A Survey on K？hler-Einstein Metrics&Shing-Tung Yau 229

Commentary on 235

Compact Three Dimensional K？hler Manifolds with Zero Ricci Curvature 235

Compact Three Dimensional K？hler Manifolds with Zero Ricci Curvature&Shing-Tung Yau 237

Commentary on 247

Inequality Between Chern Numbers of Singular K？hler Surfaces and Characterization of Orbit Space of Discrete Group of SU(2,1) 247

Inequality Between Chern Numbers of Singular K？hler Surfaces and Characterization of Orbit Space of Discrete Group of SU(2,1)&Shiu-Yuen Cheng and Shing-Tung Yau 249

Commentary on 257

On Ricci Flat 3-fold 257

On Ricci Flat 3-fold&Shi-Shyr Roan and Shing-Tung Yau 259

Commentary on 295

Some Recent Developments in General Relativity 295

Some Recent Developments in General Relativity&Shing-Tung Yau 297

Commentary on 303

Nonlinear Analysis in Geometry 303

Nonlinear Analysis in Geometry&Shing-Tung Yau 311

Commentary on 353

Three-Dimensional Algebraic Manifolds with C1＝0 and x＝-6 353

Three-Dimensional Algebraic Manifolds with C1＝0 and x＝-6&Gang Tian and Shing-Tung Yau 355

Commentary on 365

Uniformization of Geometric Structures 365

Uniformization of Geometric Structures&Shing-Tung Yau 369

Commentary on 379

Some Remarks on the Quasi-local Mass 379

Some Remarks on the Quasi-local Mass&D.Christodoulou and S.-T.Yau 381

Commentary on 385

A Review of Complex Differential Geometry 385

A Review of Complex Differential Geometry&Shing-Tung Yau 387

Commentary on 395

Open Problems in Geometry 395

Open Problems in Geometry&Shing-Tung Yau 397

Commentary on 455

On the Geometry of Certain Superconformal Field Theory Paradigms(Towards a Quantum Algebraic Geometry) 455

On the Geometry of Certain Superconformal Field Theory Paradigms(Towards a Quantum Algebraic Geometry)&Tristan Hübsch and Shing-Tung Yau 457

Commentary on 485

Upper Bound for the First Eigenvalue of Algebraic Submanifolds 485

Upper Bound for the First Eigenvalue of Algebraic Submanifolds&Jean-Pierre Bourguignon,Peter Li,and Shing-Tung Yau 487

Commentary on 495

Review on K？hler-Einstein Metrics in Algebraic Geometry 495

Review on K？hler-Einstein Metrics in Algebraic Geometry&Shing-Tung Yau 497

Commentary on 509

A Note on The Distribution of Critical Points of Eigenfunctions 509

A Note on The Distribution of Critical Points of Eigenfunctions&Shing-Tung Yau 511

Commentary on 515

Sobolev Inequality for Measure Space 515

Sobolev Inequality for Measure Space&Shing-Tung Yau 517

Commentary on 531

Einstein Manifolds with Zero Ricci Curvature 531

Einstein Manifolds with Zero Ricci Curvature&Shing-Tung Yau 533

Commentary on 549

Review of Geometry and Analysis 549

Review of Geometry and Analysis&Shing-Tung Yau 551

Commentary on 603

Geometry and Spacetime 603

Geometry and Spacetime&Shing-Tung Yau 605

Commentary on 613

Geometry Motivated by Physics 613

Geometry Motivated by Physics&Shing-Tung Yau 615

Commentary on 625

Complex Geometry:Its Brief History and Its Future 625

Complex Geometry:Its Brief History and Its Future&Shing-Tung Yau 627

Commentary on 641

Geometric Aspects of the Moduli Space of Riemann Surfaces 641

Geometric Aspects of the Moduli Space of Riemann Surfaces&Kefeng Liu,Xiaofeng Sun,and Shing-Tung Yau 643

Commentary on 671

Solution of Filtering Problem with Nonlinear Observations 671

Solution of Filtering Problem with Nonlinear Observations&Stephen S.-T.Yau and Shing-Tung Yau 673

Commentary on 699

The Proof of the Poincaré Conjecture 699

The Proof of the Poincaré Conjecture&Shing-Tung Yau 701