Handbook of geometry analysis. 几何分析手册 (第1卷) / Vol.1 =

Numerical Approximations to Extremal Metrics on Toric Surfaces .
R. S. Bunch, Simon K. Donaldson
1 Introduction
2 The set-up
3 Numerical algorithms: balanced metrics and refined approximations
4 Numerical results
5 Conclusions
References
Kahler Geometry on Toric Manifolds, and some other Manifolds with Large Symmetry
Simon K. Donaldson
Introduction
1 Background
2 Toric manifolds
3 Toric Fano manifolds
4 Variants of toric differential geometry
5 The Mukai-Umemura manifold and its deformations
References
Gluing Constructions of Special Lagrangian Cones
Mark Haskins, Nikolaos Kapouleas
I Introduction
2 Special Lagrangian cones and special Legendrian submanifolds of
S(2n-1)
3 Cohomogeneity one special Legendrian submanifolds of S(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
Jurcen Jost
1 Introduction
2 Harmonic mappings from the perspective of Riemannian geometry.
3 Harmonic mappings from the perspective of abstract analysis and convexity theory
4 Harmonic-mappings in Kahler and algebraic geometry.
5 Harmonic mappings and 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
Acknowledgement
References
Variational Principles on Triangulated Surfaces
Feng Luo
i Introduction
2 The Schlaefii formula and its counterparts in dimension 2
3 Variational principles on surfaces
4 The moduli spaces of polyhedral metrics
5 Several open problems
References
Asymptotic Structures in the Geometry of Stability and Extremal Metrics
Toshiki Mabuchi
1 Extremal metrics in Kahler 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, Joaqufn Perez, Antonio Ros
1 Introduction
2 Stability of minimal and constant mean curvature surfaces ..
3 Weak H-laminations
4 The Stable Limit Leaf Theorem
5 Foliations by constant mean curvature surfaces
6 Removable singularities and local pictures
7 Compactness of finite total curvature surfaces
8 Singular minimal laminations
9 The moduli space of embedded minimal surfaces of fixed genus
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 Kahler Manifolds
in Higher Dimensions
Luen-Fai Tam
1 Introduction
2 Function theory on KShler manifolds
3 Busemann function and the structure of nonnegatively
curved Khhler manifolds
4 Kahler-Ricci flow
5 Uniformization results
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-Ampere Eequation and its Geometric Aapplications
Neil S. Trudinger, Xu-Jia Wang
1 Introduction
2 The Monge-Ampere measure
3 A priori estimates
4 Existence and uniqueness of solutions
5 The affine metric
6 Affine maximal surfaces
References

Lectures on Mean Curvature Flows in Higher Codimensions
Mu-Tao Wang
1 Basic materials
2 Mean curvature flow
3 Blow-up analysis
4 Applications to deformations of symplectomorphisms of
Riemann surfaces
5 Acknowledgement
References

Local and Global Analysis of Eigenfunctions on Riemannian Manifolds
Steve Zelditch
Introduction
1 Basic definitions and notations
2 Explicitly solvable eigenfunctions
3 Local behavior of eigenfunctions
4 Nodal sets on C∞ Riemannian manifolds
5 The wave kernel of a compact Riemannian manifold
6 Methods for global analysis
7 Singularities pre-trace formulae
8 Weyl law and local Weyl law
9 Local and global Lp estimates of eigenfunctions
10 Gaussian beams and quasi-modes associated to
stable closed geodesics
11 Birkhoff normal forms around closed geodesics
12 Quantum integrable Laplacians
13 Concentration and non-concentration for general (M, g)
14 Lp norms and concentration in the Quantum integrable case
15 Delocalization in quantum ergodic systems, I
16 Delocalization of eigenfunctions: II: Entropy of quantum limits on manifolds with Anosov geodesic flow
17 Real analytic manifolds and their complexifieations
18 Riemannian random waves
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
4 Algebraic hyperbolicity and effective boundedness
References