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Handbook of Geometric Analysis

样章
作者：
Lizhen Ji
定价：
128.00元
ISBN：
978-7-04-025288-0
版面字数：
820千字
开本：
16开
全书页数：
676页
装帧形式：
精装
重点项目：
暂无
出版时间：
2008-08-15
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
几何学

Moduli Spaces of ProjectiveManifoldsGeometric Analysis combines differential equations anddifferential geometry. An important aspect is to solve geometricproblems by studying differential equations.Besides some knownlinear differential operators such as the Laplace operator,manydifferential equations arising from differential geometry arenonlinear. A particularly important example is the Monge-Ampreequation. Applications to geometric problems have also motivatednew methods and techniques in differential equations. The field ofgeometric analysis is broad and has had many striking applications.This handbook of geometric analysis provides introductions to andsurveys of important topics in geometric analysis and theirapplications to related fields which is intend to be referred bygraduate students and researchers in related areas.

  • 前辅文
  • Numerical Approximations to Extremal Metrics on Toric Surfaces
    • R. S. Bunch, Simon K. Donaldson.
    • 1 Introduction
    • 2 The set-up
      • 2.1 Algebraic metrics
      • 2.2 Decomposition of the curvature tensor
      • 2.3 Integration
    • 3 Numerical algorithms: balanced metrics and refined approximations
    • 4 Numerical results
      • 4.1 The hexagon
      • 4.2 The pentagon
      • 4.3 The octagon
      • 4.4 The heptagon
    • 5 Conclusions
    • References
  • Kähler Geometry on Toric Manifolds, and some other Manifolds with Large Symmetry
    • Simon K. Donaldson
    • Introduction
    • 1 Background
      • 1.1 Gauge theory and holomorphic bundles
      • 1.2 Symplectic and complex structures
      • 1.3 The equations
    • 2 Toric manifolds
      • 2.1 Local differential geometry
      • 2.2 The global structure
      • 2.3 Algebraic metrics and asymptotics
      • 2.4 Extremal metrics on toric varieties
    • 3 Toric Fano manifolds
      • 3.1 The Kähler-Ricci soliton equation
      • 3.2 Continuity method, convexity and a fundamental inequality
      • 3.3 A priori estimate
      • 3.4 The method of Wang and Zhu
    • 4 Variants of toric differential geometry
      • 4.1 Multiplicity-free manifolds
      • 4.2 Manifolds with a dense orbit
    • 5 The Mukai-Umemura manifold and its deformations
      • 5.1 Mukai's construction
      • 5.2 Topological and symplectic picture
      • 5.3 Deformations
      • 5.4 The α-invariant
    • References
  • Gluing Constructions of Special Lagrangian Cones
    • Mark Haskins, Nikolaos Kapouleas
    • 1 Introduction
    • 2 Special Lagrangian cones and special Legendrian submanifolds of □2n-1
    • 3 Cohomogeneity one special Legendrian submanifolds of □2n-1.
    • 4 Construction of the initial almost special Legendrian submanifolds
    • 5 The symmetry group and the general framework for correcting the initial surfaces
    • 6 The linearized equation
    • 7 Using the Geometric Principle to prescribe the extended substitute kernel
    • 8 The main results
    • A Symmetries and quadratics
    • References
  • Harmonic Mappings
    • Jürgen Jost
    • 1 Introduction
    • 2 Harmonic mappings from the perspective of Riemannian geometry.
      • 2.1 Harmonic mappings between Riemannian manifolds: definitions and properties
      • 2.2 The heat flow and harmonic mappings into nonpositively curved manifolds
      • 2.3 Harmonic mappings into convex regions and applications to the Bernstein problem
    • 3 Harmonic mappings from the perspective of abstract analysis and convexity theory
      • 3.1 Existence
      • 3.2 Regularity
      • 3.3 Uniqueness and some applications
    • 4 Harmonic mappings in Kähler and algebraic geometry
      • 4.1 Rigidity and superrigidity
      • 4.2 Harmonic maps and group representations
      • 4.3 Kähler groups
      • 4.4 Quasiprojective varieties and harmonic mappings of infinite energy
    • 5 Harmonic mappings and Riemann surfaces
      • 5.1 Families of Riemann surfaces
      • 5.2 Harmonic mappings from Riemann surfaces
    • References
  • Harmonic Functions on Complete Riemannian Manifolds
    • Peter Li
    • Introduction
    • 1 Gradient estimates
    • 2 Green's function and parabolicity
    • 3 Heat kernel estimates and mean value inequality
    • 4 Harmonic functions and ends
    • 5 Stability of minimal
    • 6 Polynomial growth harmonic functions
    • 7 Massive sets and the structure of harmonic maps
    • 8 Lq Harmonic functions
    • References
  • Complexity of Solutions of Partial Differential Equations
    • Fang Hua Lin
    • 1 Introduction
    • 2 Level and critical point sets
    • 3 Solutions of nonlinear equations
    • 4 A partition problem for eigenvalues
      • 4.1 Heat flow for eigenfunctions
      • 4.2 Gradient flow approach to (P)
    • Acknowledgement
    • References
  • Variational Principles on Triangulated Surfaces
    • Feng Luo
    • 1 Introduction
    • 2 The Schlaefli formula and its counterparts in dimension 2
      • 2.1 Regge calculus and Casson's approach in dimension 3
      • 2.2 The work of Colin de Verdiere, Rivin, Cohen-Kenyon-Propp and Leibon
      • 2.3 The Cosine Law and 2-dimensional Schlaefli formulas
      • 2.4 The geometric meaning of some action functionalsno
    • 3 Variational principles on surfaces
      • 3.1 Colin de Verdiere's proof of Thurston-Andreev's rigidity theorem
      • 3.2 The work of Rivin and Leibon
      • 3.3 New curvatures for polyhedral metrics and some rigidity theorems
      • 3.4 Application to Teichmiiller theory of surfaces with boundary
    • 4 The moduli spaces of polyhedral metrics
      • 4.1 Thurston-Andreev's theorem and Marden-Rodin's proof
      • 4.2 Some other results on the space of curvatures
      • 4.3 A sketch of the proof theorems 3.6 and 3.7
    • 5 Several open problems
    • References
  • Asymptotic Structures in the Geometry of Stability and Extremal Metrics Toshiki Mabuchi
    • 1 Extremal metrics in Kähler geometry
    • 2 Stability for polarized algebraic manifolds
    • 3 The asymptotic Bergman kernel
    • 4 Test configurations
    • 5 Affine sphere equations
    • 6 "Affine spheres" for toric Einstein surfaces
    • 7 Asymptotic expansion for toric Einstein surfaces
    • References
  • Stable Constant Mean Curvature Surfaces
    • William H. Meeks III, Joaquín Pérez, Antonio Ros
    • 1 Introduction
    • 2 Stability of minimal and constant mean curvature surfaces
      • 2.1 The operator ∆+q
      • 2.2 Stable H-surfaces
      • 2.3 Global theorems for stable H-surfaces
    • 3 Weak H-laminations
    • 4 The Stable Limit Leaf Theorem
    • 5 Foliations by constant mean curvature surfaces
      • 5.1 Curvature estimates and sharp mean curvature bounds for CMC foliations .
      • 5.2 Codimension one CMC foliations of □4 and □5
    • 6 Removable singularities and local pictures
      • 6.1 Structure theorems for singular CMC foliations
      • 6.2 The Local Picture Theorem on the scale of topology
      • 6.3 The statement of the theorem
    • 7 Compactness of finite total curvature surfaces
      • 7.1 The moduli space MC and the proof of Theorem 7.2
    • 8 Singular minimal laminations
    • 9 The moduli space of embedded minimal surfaces of fixed genus
      • 9.1 Conjectures on stable CMC surfaces in homogeneous three-manifolds
    • 10 Appendix
    • References
  • A General Asymptotic Decay Lemma for Elliptic Problems
    • Leon Simon
    • Introduction
    • 1 Scale invariant compact classes of submanifolds
    • 2 Some preliminaries concerning the class P
    • 3 Stability inequality
    • 4 Compact classes of cones
    • 5 A partial Harnack theory
    • 6 Proof of Theorem 1
    • 7 Application to growth estimates for exterior solutions .
    • References
  • Uniformization of Open Nonnegatively Curved Kähler Manifolds in Higher Dimensions
    • Luen-Fai Tam
    • 1 Introduction
    • 2 Function theory on Kähler manifolds
      • 2.1 Preliminary
      • 2.2 A Liouville theorem for pluri-subharmonic functions
      • 2.3 Polynomial growth holomorphic functions
    • 3 Busemann function and the structure of nonnegatively curved Kähler manifolds
      • 3.1 Curvature decay and volume growth
      • 3.2 Manifolds with nonnegative sectional curvature
    • 4 Kähler-Ricci flow
    • 5 Uniformization results
      • 5.1 Uniformization of gradient Kähler-Ricci solitons
      • 5.2 Quadratic curvature decay
      • 5.3 Linear curvature decay
    • References
  • Geometry of Measures: Harmonic Analysis Meets Geometric Measure Theory Tatiana. Toro
    • 1 Introduction
    • 2 Density-an indicator of regularity
    • 3 Harmonic measure: boundary structure and size
    • 4 Geometric measure theory tools
    • 5 Open questions
    • References
  • The Monge-Ampère Eequation and its Geometric Aapplications
    • Neil S. Trudinger, Xu-Jia Wang
    • 1 Introduction
    • 2 The Monge-Ampère measure
      • 2.1 Locally convex hypersurfaces
      • 2.2 The Monge-Ampère measure
      • 2.3 Generalized solutions
    • 3 A priori estimates
      • 3.1 Minimum ellipsoid
      • 3.2 Uniform and Hölder estimates
      • 3.3 Strict convexity and C1,α regularity
      • 3.4 Second derivative estimate
      • 3.5 C2,α estimate
      • 3.6 W2,p estimate
      • 3.7 Hölder estimate for the linearized Monge-Ampère equation
      • 3.8 Monge-Ampère equations of general form
    • 4 Existence and uniqueness of solutions
      • 4.1 The Dirichlet problem
      • 4.2 Other boundary value problems
      • 4.3 Entire solutions
      • 4.4 Hypersurfaces of prescribed Gauss curvature
      • 4.5 Variational problems for the Monge-Ampère equation
      • 4.6 Application to the isoperimetric inequality
    • 5 The affine metric
      • 5.1 Affine completeness
      • 5.2 Affine spheres
    • 6 Affine maximal surfaces
      • 6.1 The affine maximal surface equation
      • 6.2 A priori estimates
      • 6.3 The affine Bernstein problem
      • 6.4 The first boundary value problem
      • 6.5 The second boundary value problem
      • 6.6 The affine Plateau problem
    • References
  • Lectures on Mean Curvature Flows in Higher Codimensions
    • Mu-Tao Wang
    • 1 Basic materials
      • 1.1 Connections, curvature, and the Laplacian
      • 1.2 Immersed submanifolds and the second fundamental forms
      • 1.3 First variation formula
    • 2 Mean curvature flow
      • 2.1 The equation
      • 2.2 Finite time singularity
    • 3 Blow-up analysis
      • 3.1 Backward heat kernel and monotonicity formula
      • 3.2 Synopsis of singularities
    • 4 Applications to deformations of symplectomorphisms of Riemann surfaces
      • 4.1 Introduction
      • 4.2 Derivation of evolution equations
      • 4.3 Long time existence
      • 4.4 Smooth convergence as t→∞
      • 4.5 c＞0 case
      • 4.6 c=0 case
      • 4.7 c＜0 case
    • 5 Acknowledgement
    • References
  • Local and Global Analysis of Eigenfunctions on Riemannian Manifolds
    • Steve Zelditch
    • Introduction
    • 1 Basic definitions and notations
      • 1.1 Planck's constant and eigenvalue asymptotics
      • 1.2 Spectral kernels
      • 1.3 Geodesic flow
      • 1.4 Closed geodesics
      • 1.5 Jacobi fields and linear Poincaré map along a closed geodesic
      • 1.6 Geodesic flow as a unitary operator
      • 1.7 Spectrum and geodesic flow
      • 1.8 Ergodic, weak mixing and Anosov geodesic flows
      • 1.9 Completely integrable geodesic flow
      • 1.10 Quantum mechanics: wave group and pseudo-differential operators
      • 1.11 Modes and quasi-modes
      • 1.12 Heuristics and intutions
      • 1.13 Notational Index
    • 2 Explicitly solvable eigenfunctions
      • 2
      • 2.2 Flat tori
      • 2.3 Standard Sphere
      • 2.4 Surface of revolution
      • 2.5 Hn
      • 2.6 The Euclidean unit disc D
      • 2.7 An ellipse
    • 3Local behavior of eigenfunctions
      • 3.1 Eigenfuntions and harmonic functions on a cone
      • 3.2 Frequency function
      • 3.3 Doubling estimate, vanishing order estimate and lower bound estimate
      • 3.4 Semi-classical Lacunas
      • 3.5 Three ball inequalities and propagation of smallness
      • 3.6 Bernstein inequalities
      • 3.7 Carleman inequalities
      • 3.8 Geometric comparision inequalities
      • 3.9 Symmetry of positive and negative sets
      • 3.10 Alexandroff-Bakelman-Pucci-Cabré inequality
      • 3.11 Bers scaling near zeros
      • 3.12 Heuristic scaling at non-zero points
    • 4 Nodal sets on C∞ Riemannian manifolds
      • 4.1 Courant and Pleijel bounds on nodal domains
      • 4.2 Critical and singular sets of eigenfunctions on C∞ Riemannian manifolds
    • 5 The wave kernel of a compact Riemannian manifold
      • 5.1 Manifolds without conjugate points
    • 6 Methods for global analysis
      • 6.1 Egorov's theorem
      • 6.2 Sharp Garding inequality
      • 6.3 Operator norm and symbol norm
      • 6.4 Quantum Limits (Microlocal defect measures)
    • 7 Singularities pre-trace formulae
      • 7.1 Duistermaat-Guillemin short time pre-trace formula
      • 7.2 Long time pre-trace formulae
      • 7.3 Safarov trace formula
    • 8 Weyl law and local Weyl law
    • 9 Local and global Lp estimates of eigenfunctions
      • 9.1 Sketch of proof of the Sogge Lp estimate
      • 9.2 Generic non-sharpness of Sogge estimates
    • 10 Gaussian beams and quasi-modes associated to stable closed geodesics
      • 10.1 Local model
      • 10.2 WKB ansatz for a Gaussian beam
      • 10.3 Quantum Birkhoff normal form: intertwining to the model
    • 11 Birkhoff normal forms around closed geodesics
      • 11.1 Local quantum Birkhoff normal forms
      • 11.2 Model eigenfunctions around closed geodesics
    • 12 Quantum integrable Laplacian
      • 12.1 Quantum integrability and ladders of eigenfunctions
      • 12.2 Geometric examples
      • 12.3 Localization of integrable eigenfunctions
      • 12.4 Conjugation to normal form around torus orbits
    • 13 Concentration and non-concentration for general (M, g)
      • 13.1 Lp norms for restrictions to submanifolds
      • 13.2 Non-concentration in tubes around hyperbolic closed geodesics
      • 13.3 Non-concentration around closed geodesics on compact hyperbolic surfaces
    • 14 Lp norms and concentration in the Quantum integrable case
      • 14.1 Mass concentration on small length scales .
    • 15 Delocalization in quantum ergodic systems, I
      • 15.1 Quantum ergodicity in terms of operator time and space averages
      • 15.2 Quantum unique ergodicity and converse quantum ergodicity
      • 15.3 Quantum weak mixing
      • 15.4 Spectral measures and matrix elements
      • 15.5 Rate of quantum ergodicity and mixing
      • 15.6 Quantum chaos conjectures
      • 15.7 Rigorous results
      • 15.8 Quantum limits on a hyperbolic surface and Patterson-Sullivan distributions
    • 16 Delocalization of eigenfunctions: II: Entropy of quantum limits on manifolds with Anosov geodesic flow
    • 17 Real analytic manifolds and their complexifications
      • 17.1 Analytic continuation of eigenfunctions
      • 17.2 Maximal plurisubharmonic functions and growth of φ□ λ
      • 17.3 Analytic continuation and nodal hypersurfaces
      • 17.4 Nodal hypersurfaces in the case of ergodic geodesic flow
      • 17.5 Analytic domains with boundary
    • 18 Riemannian random waves
      • 18.1 Levy concentration of measure
      • 18.2 Concentration of measure and Lp norms
      • 18.3 L∞ norms: Proof of Theorem 18.3
      • 18.4 Sup norms on small balls
      • 18.5 Relation to Levy concentration
      • 18.6 Nazarov-Sodin Theorem on the mean number of nodal domains of random spherical harmonics
    • 19 Appendix on Tauberian Theorems
    • References
  • Yau's Form of Schwarz Lemma and Arakelov Inequality On Moduli Spaces of Projective Manifolds
    • Kang Zuo
    • Introduction
    • 1 Polarized complex variation of hodge structure and Higgs bundle
    • 2 Viehweg's positivity theorem for direct image sheaves
    • 3 Coverings, constructing Higgs bundles and the positivity on moduli spaces
      • 3.1 Iterated Higgs
      • 3.2 Constructing of VHS
    • 4 Algebraic hyperbolicity and effective boundedness
    • References

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