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Minimal Volume and Simplicial Volume of Visibility n-manifolds and All 3-manifolds Jianguo Cao,Xiaoyang Chen 1

1 Introduction 1

2 Simplicial Volume and Bounded Cohomology 4

3 Visibility Manifolds and Gromov-hyperbolic Spaces 6

4 Gromov's Simplicial Volume of Visibility n-manifolds and Compact 3-manifolds 10

References 12

Rigidity Theorems for Lagrangian Submanifolds of Complex Space 17

Forms with Conformal Maslov Form Xiaoli Chao,Yuxin Dong 17

1 Introduction 17

2 Preliminaries 18

3 Rigidity Theorems for Lagrangian Submanifolds 20

References 25

Method of Moving Planes in Integral Forms and Regularity Lifting Wenxiong Chen,Congming Li 27

1 Introduction 27

2 Illustration of MMP in Integral Forms 32

3 Various Applications of MMP in Integral Forms 40

4 Regularity Lifting by Contracting Operators 45

5 Regularity Lifting by Combinations of Contracting and Shrinking Operators 50

References 59

The Ricci Curvature in Finsler Geometry Xinyue Cheng 63

1 Definitions and Notations 63

2 Ricci Curvature of Randers Metrics 67

3 Volume Comparison in Finsler Geometry 70

4 The Role of the Ricci Curvature in Projective Geometry 73

References 76

Specific Non-K？hler Hermitian Metrics on Compact Complex Manifolds Jixiang Fu 79

1 Introduction 79

2 Balanced Metrics Under the Conifold Transition 80

3 The Supersymmetric Solutions 82

4 The Form-type Calabi-Yau Equation 84

5 The Generalized Gauduchon Metrics 86

References 88

On theσ2-scalar Curvature and Its Applications Yuxin Ge,Guofang Wang 91

1 Introduction 91

2 σ2-Yamabe Problem 92

3 A Quotient Equation 95

4 Ellipticity of a Quotient Equation and a 3-dimensional Sphere Theorem 98

5 A Rough Classification of Metrics of Positive Scalar Curvature 102

6 An Almost Schur Theorem 104

References 107

Isometric Embedding of Surfaces in R3 Qing Han 113

1 Introduction 113

2 Local Isometric Embedding of Surfaces 114

3 Global Isometric Embedding of Surfaces 136

References 140

The Lagrangian Mean Curvature Flow Along the K？hler-Ricci Flow Xiaoli Han,Jiayu Li 147

1 Introduction 147

2 Lagrangian Property Is Preserved 148

References 153

On Magnetohydrodynamics with Partial Magnetic Dissipation near Equilibrium Xianpeng Hu,Zhen Lei,Fang-Hua Lin 155

1 Introduction 155

2 Reformulation of the Ideal MHD 158

3 Global Existence 159

4 Acknowledgement 163

References 164

Hyperbolic Gradient Flow:Evolution of Graphs in Rn+1 De-Xing Kong,Kefeng Liu 165

1 Introduction 165

2 Hyperbolic Gradient Flow for Graphs in Rn+1 166

3 The Evolution of Convex Hypersurfaces in Rn+1 167

4 The Evolution of Plane Curves 172

5 Conclusions and Open Problems 176

References 177

The Moser-Trudinger and Adams Inequalities and Elliptic and Subelliptic Equations with Nonlinearity of Exponential Growth Nguyen Lam,Guozhen Lu 179

1 Introduction 180

2 The Moser-Trudinger Inequalities 183

3 Adams Type Inequalities on High Order Sobolev Spaces 189

4 N-Laplacian on Bounded Domains in RN 193

5 Polyharmonic Equations on Bounded Domains in R2m 199

6 N-Laplacian Equations with Critical Exponential Growth in RN 202

7 Existence of Solutions to Polyharmonic Equations with Critical Exponential Growth in the Whole Space 215

8 Subcritical Exponential Growth in Bounded Domains 221

9 Equations of Q-sub-Laplacian Type with Critical Growth on Bounded Domains 226

10 Equations of Q-sub-Laplacian Type with Critical Growth in Hn 232

11 Examples of Nonlinear Terms Without the Ambrosetti-Rabinowitz Condition 236

12 Sharp Moser-Trudinger Inequality on the Heisenberg Group at the Critical Case and Multiplicity of Solutions 237

13 A New Approach to Sharp Adams Inequalities for Arbitrary Integers and Less Restrictive Norms 243

References 246

Navigation Problem and Randers Metrics Meng Li,Xiaohuan Mo,Ni Yu 253

1 Historical Remarks and Definitions 253

2 Navigation Problem 255

3 Flag Curvature Decreasing Property of Navigation Problem 257

4 New Finsler Metrics with Constant(Scalar)Flag Curvature from Old 258

5 Geodesics of a Finsler Metric Via Navigation Problem 260

6 A New Characterization of Randers Norms 260

7 Compact Randers Metrics with Constant S-curvature 261

8 Classification Results for Randers Metrics 263

9 Construction of Randers Metrics with Isotropic S-curvature 264

References 265

Lorentzian Isoparametric Hypersurfaces in the Lorentzian Sphere Sn1+1 Zhen-Qi Li 267

1 Introduction 267

2 Canonical Forms for Symmetric Transformation A in Lorentzian Space 269

3 Lorentzian Isoparametric Hypersurface of Type I in Sn1+1 273

4 Nonexistence of Lorentzian Isoparametric Hypersurface of Type Ⅳ in Sn1+1 279

5 Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1 283

6 Totally Umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1 291

7 Semi-umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1 304

8 Non-umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1 310

References 328

Asymptotically Hyperbolic Manifolds and Conformal Geometry Jie Qing 329

1 Definitions 329

2 Regularity and Rigidity for AH Manifold 332

3 Correspondences 333

4 Generalized Yamabe Problems 338

References 342

A Characterization of Randers Metrics of Scalar Flag Curvature Zhongmin Shen…Gulcin Civi 345

1 Introduction 345

2 Preliminaries 348

3 A Formula for the Weyl Curvature 350

4 Proof of Theorem 1.1 351

5 Weak Einstein Metrics 354

6 Proof of Theorem 1.2 355

References 357

Prescribed Weingarten Curvature Equations Weimin Sheng,Neil S.Trudinger,Xu-Jia Wang 359

1 Introduction 359

2 The Uniform Estimate 363

3 The Gradient Estimates 366

4 The Second Order Derivative Estimate 371

5 The Dirichlet Problem 376

6 Closed Convex Hypersurfaces 378

References 383

Geometry Problems Related with Quasi-local Mass in General Relativity Yuguang Shi 387

1 Introduction 387

2 Quasi-local Mass on Domains with Convex Boundaries 389

3 Quasi-local Mass on Domains with Non Convex Boundarv 402

References 404

Concerning the L4 Norms of Typical Eigenfunctions on Compact Surfaces Christopher D.Sogge,Steve Zelditch 407

1 Introduction 407

2 L4 Norms of Generic Eigenfunctions 411

3 Average L4 Norms of Spherical Harmonics 418

4 L4 Norms of Orthonormal Bases of Spherical Harmonics 419

5 Random Orthonormal Bases of Spherical Harmonics 419

6 Other Orthonormal Bases 421

References 421

Analysis on Riemannian Manifolds with Non-convex Boundarv Feng-Yu Wang 425

1 Introduction 425

2 Semigroup Properties for Curvature and Second Fundamental Form 426

3 Dimension-free Harnack Inequalities 428

4 Functional Inequalities 431

References 437

The Unity of pharmonic Geometrv Shihshu Walter Wei 439

1 Introduction 439

2 A Perspective ofp-harmonic Maps 442

3 p-Harmonic Maps and Cohomology Groups 455

4 Unified Notions of F-harmonic Maps 457

5 Unified Notions of F-Yang-Mills Fields 458

6 Vector Bundle Valued p-forms on Riemannian Manifolds:A Higher Level of Generalization 460

7 Manifolds with the Global Doubling Property in p-harmonic Geometry 464

8 Generalized 1-harmonic Equations and Geometric Flows 465

9 The Mean Curvature Flow 470

10 Generalized Constant Mean Curvature Type Equations for Differential Forms 472

11 Geometric Inequalities on Manifolds 473

12 Harmonic Maps in Finsler Geometry 476

References 476

Hermitian Harmonic Maps Between Almost Hermitian Manifolds Xi Zhang 485

1 Introduction 485

2 Preliminary Results 486

3 Proof of Theorem 1.1 490

References 492

A Global Mean Value Inequality for Plurisubharmonic Functions on a Compact K？ihler Manifold Xiaohua Zhu 495

1 Introduction 495

2 Relative Capacity and L∞-estimate 497

3 Proof of Theorem 1.1 500

4 A Discussion on Theorem 1.1 504

References 506

A List of Publications by Professor Zhengguo Bai 509

A List of Publications by Professor Yibing Shen 511

A List of Graduate Students of Professors Bai and Shen 517