幾何分析將微分幾何和微分方程聯(lián)系在一起，其中最主要的特征就是通過研究微分方程解決幾何問題。幾何分析的領(lǐng)域廣泛且有著許多重要的應(yīng)用。本書收集了16篇在國際幾何分析界知名教授為此次會議專門準備的報告論文，包含幾何分析各個方面的最新進展。
《Handbook of Geometric Analysis》講述了Geometric Analysis combines difierentiaI equations and difierential geometry．Animportant aspect iS to solve geometric problems by studying difierentiaI equations．Besides some known linear difierential operators such as the Laplace operator,many difierential equations arising from difrerential geometry are nonlinear．Aparticularly important example iS the Monge-Amp~~re equation。Applications togeometric problems have also motivated new methods and techniques in differen-tiaI equations。The field of geometric analysis iS broad and has had many strikingapplications．This handbook of geometric analysis provides introductions to andsurveys of important topics in geometric analysis and their applications to relatedfields which iS intend t0 be referred by graduate students and researchers in relatedareas．
Numerical Approximations to Extremal Metrics on Toric Surfaces
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
Kahler Geometry on Toric Manifolds, and some other Manifolds with Large Symmetry
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 Kahler-Ricci soliton equation
3.2 Continuity method, convexity and a fundamentalinequality
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 a-invariant
References
Gluing Constructions of Special Lagrangian Cones
1Introduction
2 Special Lagrangian cones and special Legendrian submanifolds of S2n-1
3 Cohomogeneity one special Legendrian submanifolds of S2n-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
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 Kahler and algebraic geometry
4.1 Rigidity and superrigidity
4.2 Harmonic maps and group representations
4.3 Kahler groups
4.4 Quasiprojective varieties and harmonic mappings of infinite energy
5 Harmonic mappings and Riemann surfaces
5.1 Families of Riemann surfaces
……
Harmonic Functions on Complete Riemannian Manifolds
Complexity of Solutions of Partial Differential Equations
Variational Principles on Triangulated Surfaces
Asymptotic Structures in the Geometry of Stability and Extremal Metrics
Stable Constant Mean Curvature Surfaces
A General Asymptotic Decay Lemma for Elliptic Problems
Uniformization of Open Nonnegatively Curved K／ihler Manifolds in Higher Dimensions
Geometry of Measures：Harmonic Analysis Meets Geometric Measure Theory
The Monge Ampere Eequation and its Geometric Aapplications
Lectures on Mean Curvature Flows in Higher Codimensions
Local and Global Analysis of Eigenfunctions on Riemannian Manifolds
Yau’S Form of Schwarz Lemma and Arakelov Inequality On Moduli Spaces of Projective Manifolds
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