几何分析手册 第2卷 英文版PDF|Epub|txt|kindle电子书版本网盘下载

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Heat Kernels on Metric Measure Spaces with Regular Volume Growth&Alexander Grigor'yan1

1 Introduction1

1.1 Heat kernel in Rn2

1.2 Heat kernels on Riemannian manifolds3

1.3 Heat kernels of fractional powers of Laplacian4

1.4 Heat kernels on fractal spaces5

1.5 Summary of examples7

2 Abstract heat kernels8

2.1 Basic definitions8

2.2 The Dirichlet form11

2.3 Identifying Ф in the non-local case13

2.4 Volume of balls17

3 Besov spaces21

3.1 Besov spaces in Rn21

3.2 Besov spaces in a metric measure space23

3.3 Embedding of Besov spaces into H？lder spaces24

4 The energy domain26

4.1 A local case26

4.2 Non-local case31

4.3 Subordinated heat kernel32

4.4 Bessel potential spaces35

5 The walk dimension36

5.1 Intrinsic characterization of the walk dimension36

5.2 Inequalities for the walk dimension39

6 Two-sided estimates in the local case46

6.1 The Dirichlet form in subsets46

6.2 Maximum principles47

6.3 A tail estimate47

6.4 Identifying Ф in the local case55

References57

A Convexity Theorem and Reduced Delzant Spaces&Bong H.Lian,Bailin Song61

1 Introduction61

2 Convexity of image of moment map64

3 Rationality of moment polytope69

4 Realizing reduced Delzant spaces74

5 Classification of reduced Delzant spaces82

References94

Localization and some Recent Applications&Bong H.Lian,Kefeng Liu97

1 Introduction97

2 Localization100

3 Mirror principle102

4 Hori-Vafa formula112

5 The Mari？o-Vafa Conjecture115

6 Two partition formula123

7 Theory of topological vertex125

8 Gopakumar-Vafa conjecture and indices of elliptic operators128

9 Two proofs of the ELSV formula129

10 A localization proof of the Witten conjecture132

11 Final remarks134

References134

Gromov-Witten Invariants of Toric Calabi-Yau Threefolds&Chiu-Chu Melissa Liu139

1 Gromov-Witten invariants of Calabi-Yau 3-folds139

1.1 Symplectic and algebraic Gromov-Witten invariants139

1.2 Moduli space of stable maps139

1.3 Gromov-Witten invariants of compact Calabi-Yau 3-folds140

1.4 Gromov-Witten invariants of noncompact Calabi-Yau 3-folds141

2 Traditional algorithm in the toric case142

2.1 Localization142

2.2 Hodge integrals143

3 Physical theory of the topological vertex144

4 Mathematical theory of the topological vertex146

4.1 Locally planar trivalent graph146

4.2 Formal toric Calabi-Yau(FTCY)graphs148

4.3 Degeneration formula150

4.4 Topological vertex152

4.5 Localization153

4.6 Framing dependence154

4.7 Combinatorial expression154

4.8 Applications155

4.9 Comparison155

5 GW/DT correspondences and the topological vertex156

Acknowledgments156

References156

Survey on Affine Spheres&John Loftin161

1 Introduction161

2 Affine structure equations163

3 Examples164

4 Two-dimensional affine spheres and Titeica's equation165

5 Monge-Ampère equations and duality168

6 Global classification of affine spheres172

7 Hyperbolic affine spheres and invariants of convex cones173

8 Projective manifolds176

9 Affinc manifolds181

10 Affine maximal hypersurfaces185

11 Affine normal flow186

References187

Convergence and Collapsing Theorems in Riemannian Geometry&Xiaochun Rong193

Introduction193

1 Gromov-Hausdorff distance in space of metric spaces194

1.1 The Gronov-Hausdorff distance194

1.2 Examples199

1.3 An alternative formulation of GH-distance202

1.4 Compact subsets of(Met,dGH)204

1.5 Equivariant GH-convergence206

1.6 Pointed GH-convergence209

2 Smooth limits-fibrations217

2.1 The fibration theorem217

2.2 Sectional curvature comparison219

2.3 Embedding via distance functions223

2.4 Fibrations226

2.5 Proof of theorem 2.1.1231

2.6 Center of mass234

2.7 Equivariant fibrations235

2.8 Applieations of the fibration theorem240

3 Convergence theorems245

3.1 Cheeger-Gromov's convergence theorem245

3.2 Injectivity radius estimate248

3.3 Some elliptic estimates253

3.4 Harmonic radius estimate255

3.5 Smoothing metrics259

4 Singular limits-singular fibrations260

4.1 Singular fibrations261

4.2 Controlled homotopy structure by geometry265

4.3 The π2-finiteness theorem269

4.4 Collapsed manifolds with pinched positive sectional curvature271

5 Almost flat manifolds273

5.1 Gromov's theorem on almost flat manifolds273

5.2 The Margulis lemma275

5.3 Flat connections with small torsion277

5.4 Flat connection with a parallel torsion281

5.5 Proofs—part Ⅰ285

5.6 Proofs—part Ⅱ290

5.7 Refined fibration theorem294

References297

Geometric Transformations and Soliton Equations&Chuu-Lian Terng301

1 Introduction301

2 The moving frame method for submanifolds306

3 Line congruences and B？cklund transforms309

4 Sphere congruences and Ribaucour transforms315

5 Combescure transforms,O-surfaces,and k-tuples317

6 From moving frame to Lax pair320

7 Soliton hierarchies constructed from symmetric spaces329

8 The U/K-system and the Gauss-Codazzi equations336

9 Loop group actions343

10 Action of simple elements and geometric transforms347

References355

Affine Integral Geometry from a Differentiable Viewpoint&Deane Yang359

1 Introduction359

2 Basic definitions and notation361

2.1 Linear group actions361

3 Objects of study362

3.1 Geometric setting362

3.2 Convex body362

3.3 The space of all convex bodies362

3.4 Valuations362

4 Overall strategy363

5 Fundamental constructions363

5.1 The support function363

5.2 The Minkowski sum364

5.3 The polar body365

5.4 The inverse Gauss map366

5.5 The second fundamental form366

5.6 The Legendre transform366

5.7 The curvature function367

6 The homogeneous contour integral368

6.1 Homogeneous functions and differential forms368

6.2 The homogeneous contour integral for a differential form369

6.3 The homogeneous contour integral for a measure369

6.4 Homogeneous integral calculus373

7 An explicit construction of valuations374

7.1 Duality375

7.2 Volume375

8 Classification of valuations376

9 Scalar valuations376

9.1 SL(n)-invariant valuations376

9.2 Hug's theorem378

10 Continuous GL(n)-homogeneous valuations378

10.1 Scalar valuations378

10.2 Vector-valued valuations379

11 Matrix-valued valuations380

11.1 The Cramer-Rao inequality381

12 Homogeneous function-and convex body-valued valuations383

13 Questions384

References385

Classification of Fake Projective Planes&Sai-Kee Yeung391

1 Introduction391

2 Uniformization of fake projective planes393

3 Geometric estimates on the number of fake projective planes396

4 Arithmeticity of lattices associated to fake projective planes398

5 Covolume formula of Prasad410

6 Formulation of proof411

7 Statements of the results419

8 Further studies423

References427