作者簡介
編者：（美國）季理真等
圖書目錄
Heat Kernels on Metric Measure Spaces with Regu
幾何分析手冊lar 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
