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Minimal Volume and Simplicial Volume of Visibility n-manifolds and All 3-manifolds Jianguo Cao,Xiaoyang Chen1

1 Introduction1

2 Simplicial Volume and Bounded Cohomology4

3 Visibility Manifolds and Gromov-hyperbolic Spaces6

4 Gromov's Simplicial Volume of Visibility n-manifolds and Compact 3-manifolds10

References12

Rigidity Theorems for Lagrangian Submanifolds of Complex Space17

Forms with Conformal Maslov Form Xiaoli Chao,Yuxin Dong17

1 Introduction17

2 Preliminaries18

3 Rigidity Theorems for Lagrangian Submanifolds20

References25

Method of Moving Planes in Integral Forms and Regularity Lifting Wenxiong Chen,Congming Li27

1 Introduction27

2 Illustration of MMP in Integral Forms32

3 Various Applications of MMP in Integral Forms40

4 Regularity Lifting by Contracting Operators45

5 Regularity Lifting by Combinations of Contracting and Shrinking Operators50

References59

The Ricci Curvature in Finsler Geometry Xinyue Cheng63

1 Definitions and Notations63

2 Ricci Curvature of Randers Metrics67

3 Volume Comparison in Finsler Geometry70

4 The Role of the Ricci Curvature in Projective Geometry73

References76

Specific Non-K？hler Hermitian Metrics on Compact Complex Manifolds Jixiang Fu79

1 Introduction79

2 Balanced Metrics Under the Conifold Transition80

3 The Supersymmetric Solutions82

4 The Form-type Calabi-Yau Equation84

5 The Generalized Gauduchon Metrics86

References88

On theσ2-scalar Curvature and Its Applications Yuxin Ge,Guofang Wang91

1 Introduction91

2 σ2-Yamabe Problem92

3 A Quotient Equation95

4 Ellipticity of a Quotient Equation and a 3-dimensional Sphere Theorem98

5 A Rough Classification of Metrics of Positive Scalar Curvature102

6 An Almost Schur Theorem104

References107

Isometric Embedding of Surfaces in R3 Qing Han113

1 Introduction113

2 Local Isometric Embedding of Surfaces114

3 Global Isometric Embedding of Surfaces136

References140

The Lagrangian Mean Curvature Flow Along the K？hler-Ricci Flow Xiaoli Han,Jiayu Li147

1 Introduction147

2 Lagrangian Property Is Preserved148

References153

On Magnetohydrodynamics with Partial Magnetic Dissipation near Equilibrium Xianpeng Hu,Zhen Lei,Fang-Hua Lin155

1 Introduction155

2 Reformulation of the Ideal MHD158

3 Global Existence159

4 Acknowledgement163

References164

Hyperbolic Gradient Flow:Evolution of Graphs in Rn+1 De-Xing Kong,Kefeng Liu165

1 Introduction165

2 Hyperbolic Gradient Flow for Graphs in Rn+1166

3 The Evolution of Convex Hypersurfaces in Rn+1167

4 The Evolution of Plane Curves172

5 Conclusions and Open Problems176

References177

The Moser-Trudinger and Adams Inequalities and Elliptic and Subelliptic Equations with Nonlinearity of Exponential Growth Nguyen Lam,Guozhen Lu179

1 Introduction180

2 The Moser-Trudinger Inequalities183

3 Adams Type Inequalities on High Order Sobolev Spaces189

4 N-Laplacian on Bounded Domains in RN193

5 Polyharmonic Equations on Bounded Domains in R2m199

6 N-Laplacian Equations with Critical Exponential Growth in RN202

7 Existence of Solutions to Polyharmonic Equations with Critical Exponential Growth in the Whole Space215

8 Subcritical Exponential Growth in Bounded Domains221

9 Equations of Q-sub-Laplacian Type with Critical Growth on Bounded Domains226

10 Equations of Q-sub-Laplacian Type with Critical Growth in Hn232

11 Examples of Nonlinear Terms Without the Ambrosetti-Rabinowitz Condition236

12 Sharp Moser-Trudinger Inequality on the Heisenberg Group at the Critical Case and Multiplicity of Solutions237

13 A New Approach to Sharp Adams Inequalities for Arbitrary Integers and Less Restrictive Norms243

References246

Navigation Problem and Randers Metrics Meng Li,Xiaohuan Mo,Ni Yu253

1 Historical Remarks and Definitions253

2 Navigation Problem255

3 Flag Curvature Decreasing Property of Navigation Problem257

4 New Finsler Metrics with Constant(Scalar)Flag Curvature from Old258

5 Geodesics of a Finsler Metric Via Navigation Problem260

6 A New Characterization of Randers Norms260

7 Compact Randers Metrics with Constant S-curvature261

8 Classification Results for Randers Metrics263

9 Construction of Randers Metrics with Isotropic S-curvature264

References265

Lorentzian Isoparametric Hypersurfaces in the Lorentzian Sphere Sn1+1 Zhen-Qi Li267

1 Introduction267

2 Canonical Forms for Symmetric Transformation A in Lorentzian Space269

3 Lorentzian Isoparametric Hypersurface of Type I in Sn1+1273

4 Nonexistence of Lorentzian Isoparametric Hypersurface of Type Ⅳ in Sn1+1279

5 Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1283

6 Totally Umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1291

7 Semi-umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1304

8 Non-umbilical Lorentzian Isoparametric Hypersurface of Type Ⅱ in Sn1+1310

References328

Asymptotically Hyperbolic Manifolds and Conformal Geometry Jie Qing329

1 Definitions329

2 Regularity and Rigidity for AH Manifold332

3 Correspondences333

4 Generalized Yamabe Problems338

References342

A Characterization of Randers Metrics of Scalar Flag Curvature Zhongmin Shen…Gulcin Civi345

1 Introduction345

2 Preliminaries348

3 A Formula for the Weyl Curvature350

4 Proof of Theorem 1.1351

5 Weak Einstein Metrics354

6 Proof of Theorem 1.2355

References357

Prescribed Weingarten Curvature Equations Weimin Sheng,Neil S.Trudinger,Xu-Jia Wang359

1 Introduction359

2 The Uniform Estimate363

3 The Gradient Estimates366

4 The Second Order Derivative Estimate371

5 The Dirichlet Problem376

6 Closed Convex Hypersurfaces378

References383

Geometry Problems Related with Quasi-local Mass in General Relativity Yuguang Shi387

1 Introduction387

2 Quasi-local Mass on Domains with Convex Boundaries389

3 Quasi-local Mass on Domains with Non Convex Boundarv402

References404

Concerning the L4 Norms of Typical Eigenfunctions on Compact Surfaces Christopher D.Sogge,Steve Zelditch407

1 Introduction407

2 L4 Norms of Generic Eigenfunctions411

3 Average L4 Norms of Spherical Harmonics418

4 L4 Norms of Orthonormal Bases of Spherical Harmonics419

5 Random Orthonormal Bases of Spherical Harmonics419

6 Other Orthonormal Bases421

References421

Analysis on Riemannian Manifolds with Non-convex Boundarv Feng-Yu Wang425

1 Introduction425

2 Semigroup Properties for Curvature and Second Fundamental Form426

3 Dimension-free Harnack Inequalities428

4 Functional Inequalities431

References437

The Unity of pharmonic Geometrv Shihshu Walter Wei439

1 Introduction439

2 A Perspective ofp-harmonic Maps442

3 p-Harmonic Maps and Cohomology Groups455

4 Unified Notions of F-harmonic Maps457

5 Unified Notions of F-Yang-Mills Fields458

6 Vector Bundle Valued p-forms on Riemannian Manifolds:A Higher Level of Generalization460

7 Manifolds with the Global Doubling Property in p-harmonic Geometry464

8 Generalized 1-harmonic Equations and Geometric Flows465

9 The Mean Curvature Flow470

10 Generalized Constant Mean Curvature Type Equations for Differential Forms472

11 Geometric Inequalities on Manifolds473

12 Harmonic Maps in Finsler Geometry476

References476

Hermitian Harmonic Maps Between Almost Hermitian Manifolds Xi Zhang485

1 Introduction485

2 Preliminary Results486

3 Proof of Theorem 1.1490

References492

A Global Mean Value Inequality for Plurisubharmonic Functions on a Compact K？ihler Manifold Xiaohua Zhu495

1 Introduction495

2 Relative Capacity and L∞-estimate497

3 Proof of Theorem 1.1500

4 A Discussion on Theorem 1.1504

References506

A List of Publications by Professor Zhengguo Bai509

A List of Publications by Professor Yibing Shen511

A List of Graduate Students of Professors Bai and Shen517