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内容简介

作者简介

图书目录

图书信息

书 名: 几何分析手册

作　者：季理真

出版社： 高等教育出版社

出版时间： 2010年4月1日

ISBN: 9787040288834

开本： 16开

定价: 78.00元

内容简介

《几何分析手册(第2卷)》内容简介：The marriage of geometry and analysis, in particular non-linear differential equations, has been very fruitful. An early deep application of geometric analysis is the celebrated solution by Shing-Tung Yau of the Calabi conjecture in 1976. In fact, Yau together with many of his collaborators developed important techniques in geometric analysis in order to solve the Calabi conjecture.

作者简介

编者：（美国）季理真 等

图书目录

Heat Kernels on Metric Measure Spaces with Regular Volume Growth

Alexander Griqor'yan

1 Introduction

1.1 Heat kernel in Rn

1.2 Heat kernels on Riemannian manifolds

1.3 Heat kernels of fractional powers of Laplacian

1.4 Heat kernels on fractal spaces

1.5 Summary of examples

2 Abstract heat kernels

2.1 Basic definitions

2.2 The Dirichlet form

2.3 Identifying in the non-local case

2.4 Volume of balls

3 Besov spaces

3.1 Besov spaces in Rn

3.2 Besov spaces in a metric measure space

3.3 Embedding of Besov spaces into HSlder spaces.

4 The energy domain

4.1 A local case

4.2 Non-local case

4.3 Subordinated heat kernel

4.4 Bessel potential spaces

5 The walk dimension

5.1 Intrinsic characterization of the walk dimension

5.2 Inequalities for the walk dimension

6 Two-sided estimates in the local case

6.1 The Dirichlet form in subsets

6.2 Maximum principles

6.3 A tail estimate

6.4 Identifying in the local case

References

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

1 Introduction

2 Convexity of image of moment map

3 Rationality of moment polytope

4 Realizing reduced Delzant spaces

5 Classification of reduced Delzant spaces

References

Localization and some Recent Applications

Bong H. Lian, Kefeng Liu

1 Introduction

2 Localization

3 Mirror principle

4 Hori-Vafa formula

5 The Marino-Vafa Conjecture

6 Two partition formula

7 Theory of topological vertex

8 Gopakumar-Vafa conjecture and indices of elliptic operators..

9 Two proofs of the ELSV formula

10 A localization proof of the Witten conjecture

11 Final remarks

References

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

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

1.1 Symplectic and algebraic Gromov-Witten invariants

1.2 Moduli space of stable maps

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

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

2 Traditional algorithm in the toric case

2.1 Localization

2.2 Hodge integrals

3 Physical theory of the topological vertex

4 Mathematical theory of the topological vertex

4.1 Locally planar trivalent graph

4.2 Formal toric Calabi-Yau (FTCY) graphs

4.3 Degeneration formula

4.4 Topological vertex "

4.5 Localization

4.6 Framing dependence

4.7 Combinatorial expression

4.8 Applications

4.9 Comparison

5 GW/DT correspondences and the topological vertex

Acknowledgments

References

Survey on Affine Spheres

John Loftin

1 Introduction

2 Affine structure equations

3 Examples

4 Two-dimensional affine spheres and Titeica's equation

5 Monge-Ampre equations and duality

6 Global classification of affine spheres

7 Hyperbolic affine spheres and invariants of convex cones

8 Projective manifolds

9 Affine manifolds

10 Affine maximal hypersurfaces

11 Affine normal flow

References

Convergence and Collapsing Theorems in Riemannian Geometry

Xiaochun Rong

Introduction

1 Gromov-Hausdorff distance in space of metric spaces

1.1 The Gromov-Hausdorff distance

1.2 Examples

1.3 An alternative formulation of GH-distance

1.4 Compact subsets of (Met, dGH)

1.5 Equivariant GH-convergence

1.6 Pointed GH-convergence

2 Smooth limits-fibrations

2.1 The fibration theorem

2.2 Sectional curvature comparison

2.3 Embedding via distance functions

2.4 Fibrations

2.5 Proof of theorem 2.1.1

2.6 Center of mass

2.7 Equivariant fibrations

2.8 Applications of the fibration theorem

3 Convergence theorems

3.1 Cheeger-Gromov's convergence theorem

3.2 Injectivity radius estimate

3.3 Some elliptic estimates

3.4 Harmonic radius estimate

3.5 Smoothing metrics

4 Singular limits-singular fibrations

4.1 Singular fibrations

4.2 Controlled homotopy structure by geometry

4.3 The ∏2-finiteness theorem

4.4 Collapsed manifolds with pinched positive sectional curvature

5 Almost flat manifolds

5.1 Gromov's theorem on almost flat manifolds

5.2 The Margulis lemma

5.3 Flat connections with small torsion

5.4 Flat connection with a parallel torsion

5.5 Proofs——part I

5.6 Proofs——part II

5.7 Refined fibration theorem

References

Geometric Transformations and Soliton Equations

Chuu-Lian Terng "

1 Introduction

2 The moving frame method for submanifolds

3 Line congruences and Backlund transforms

4 Sphere congruences and Ribaucour transforms

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

6 From moving frame to Lax pair

7 Soliton hierarchies constructed from symmetric spaces

8 The U-system and the Gauss-Codazzi equations

9 Loop group actions

10 Action of simple elements and geometric transforms

References

Affine Integral Geometry from a Differentiable Viewpoint

Deane Yang

1 Introduction

2 Basic definitions and notation

2.1 Linear group actions

3 Objects of study

3.1 Geometric setting

3.2 Convex body

3.3 The space of all convex bodies

3.4 Valuations

4 Overall strategy

5 Fundamental constructions

5.1 The support function

5.3 The polar body

5.4 The inverse Gauss map

5.5 The second fundamental form

5.6 The Legendre transform

5.7 The curvature function The homogeneous contour integral

6.1 Homogeneous functions and differential forms

6.2 The homogeneous contour integral for a differential form

6.3 The homogeneous contour integral for a measure

6.4 Homogeneous integral calculus

7 An explicit construction of valuations

7.1 Duality

7.2 Volume

8 Classification of valuations

9 Scalar valuations

9.1 SL(n)-invariant valuations

9.2 Hug's theorem

10 Continuous GL(n)-homogeneous valuations

10.1 Scalar valuations

10.2 Vector-valued valuations

11 Matrix-valued valuations.

11.1 The Cramer-Rao inequality

12 Homogeneous function- and convex body-valued valuations.

13 Questions

References

Classification of Fake Projective Planes

Sai-Kee Yeung

1 Introduction

2 Uniformization of fake projective planes

3 Geometric estimates on the number of fake projective planes.

4 Arithmeticity of lattices associated to fake projective planes.

5 Covolume formula of Prasad

6 Formulation of proof

7 Statements of the results

8 Further studies

References