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.

Geometric Analysis combines differential equations and differential geometry. Animportant aspect is to solve geometric problems by studying differential equations.Besides some known linear differential operators such as the Laplace operator,many differential equations arising from differential geometry are nonlinear. Aparticularly important example is the Monge-Ampere equation. Applications togeometric problems have also motivated new methods and techniques in differen-tial 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 to be referred by graduate students and researchers in relatedareas.

A Survey of Einstein Metrics on 4-manifolds

 Michael T. Anderson

 1 Introduction

 2 Brief review： 4-manifolds, complex surfaces and Einstein metrics

 3 Constructions of Einstein metrics Ⅰ

 4 Obstructions to Einstein metrics

 5 Moduli spaces Ⅰ

 6 Moduli spaces Ⅱ

 7 Constructions of Einstein metrics Ⅱ

 8 Concluding remarks

 References

Sphere Theorems in Geometry

 Simon Brendle, Richard Schoen

 1 The Topological Sphere Theorem

 2 Manifolds with positive isotropic curvature

 3 The Differentiable Sphere Theorem

 4 New invariant curvature conditions for the Ricci flow

 5 Rigidity results and the classification of weakly 1/4-pinched manifolds

 6 Hamilton's differential Harnack inequality for the Ricci flow

 7 Compactness of pointwise pinched manifolds

 References

Curvature Flows and CMC Hypersurfaces

 Claus Gerhardt

 1 Introduction

 2 Notations and preliminary results

 3 Evolution equations for some geometric quantities

 4 Essential parabolic flow equation

 5 Existence results

 6 Curvature flows in Riemannian manifolds

 7 Foliation of a spacetime by CMC hypersurfaces

 8 The inverse mean curvature flow in Lorentzian spaces

 References

Geometric Structures on Riemannian Manifolds

 Naichung Conan Leung

 1 Introduction

 2 Topology of manifolds

 2.1 Cohomology and geometry of differential forms

 2.2 Hodge theorem

 2.3 Witten-Morse theory

 2.4 Vector bundles and gauge theory

 3 Riemannian geometry

 3.1 Torsion and Levi-Civita connections

 3.2 Classification of Riemannian holonomy groups

 3.3 Riemannian curvature tensors

 3.4 Flat tori

 3.5 Einstein metrics

 3.6 Minimal submanifolds

 3.7 Harmonic maps

 4 Oriented four manifolds

 4.1 Gauge theory in dimension four

 4.2 Riemannian geometry in dimension four

 5 Kahler geometry

 5.1 Kahler geometry — complex aspects

 5.2 Kahler geometry — Riemannian aspects

 5.3 Kahler geometry — symplectic aspects

 5.4 Gromov-Witten theory

 6 Calabi-Yau geometry

 6.1 Calabi-Yau manifolds

 6.2 Special Lagrangian geometry

 6.3 Mirror symmetry

 6.4 K3 surfaces

 7 Calabi-Yau 3-folds

 7.1 Moduli of CY threefolds

 7.2 Curves and surfaces in Calabi-Yau threefolds

 7.3 Donaldson-Thomas bundles over Calabi-Yau threefolds...

 7.4 Special Lagrangian submanifol'ds in CY3

 7.5 Mirror symmetry for Calabi-Yau threefolds

 8 G2-geometry

 8.1 G2-manifolds

 8.2 Moduli of G2-manifolds

 8.3 （Co-）associative geometry

 8.4 G2-Donaldson-Thomas bundles

 8.5 G2-dualities, trialities and M-theory

 9 Geometry of vector cross products

 9.1 VCP manifolds

 9.2 Instantons and branes

 9.3 Symplectic geometry on higher dimensional knot spaces

 9.4 C-VCP geometry

 9.5 Hyperkahler geometry on isotropic knot spaces of CY

 10 Geometry over normed division algebras

 I0.1 Manifolds over normed algebras

 10.2 Gauge theory over （special） A-manifolds

 10.3 A-submanifolds and （special） Lagrangian submanifolds

 11 Quaternion geometry

 11.1 Hyperkahler geometry

 11.2 Quaternionic-Kahler geometry

 12 Conformal geometry

 13 Geometry of Riemannian symmetric spaces

 13.1 Riemannian symmetric spaces

 13.2 Jordan algebras and magic square

 13.3 Hermitian and quaternionic symmetric spaces

 14 Conclusions

 References

Symplectic Calabi-Yau Surfaces

 Tian-Jun Li

 1 Introduction

 2 Linear symplectic geometry

 2.1 Symplectic vector spaces

 2.2 Compatible complex structures

 2.3 Hermitian vector spaces

 2.4 4-dimensional geometry

 3 Symplectic manifolds

 3,1 Almost symplectic and almost complex structures

 3.2 Cohomological invariants and space of symplectic structures

 3.3 Moser stability and Darboux charts

 3.4 Submanifolds and their neighborhoods

 3.5 Constructions

 4 Almost Kahler geometry

 4.1 Almost Hermitian manifolds, integrability and operators

 4.2 Levi-Civita connection

 4.3 Connections and curvature on Hermitian bundles

 4.4 Chern connection and Hermitian curvatures

 4.5 The self-dual operator

 4.6 Dirac operators

 4.7 Weitzenbock formulas and some almost Kahler identities

 5 Seiberg-Witten theory-three facets

 5.1 SW equations

 5.2 Pin（2） symmetry for a spin reduction

 5.3 The compactness and Hausdorff property of the moduli space

 5.4 Generic smoothness of the moduli space

 5.5 Furuta's finite dim. Approximations

 5.6 SW invariants

 5.7 Symplectic SW equa.tions and Taubes' nonvanishing

 5.8 Symplectic SW solutions and Pseudo-holomorphic curves

 5.9 Bordism SW invariants via finite dim. Approximations

 5.10 Mod 2 vanishing and homology type

 6 Symplectic Calabi-Yau equation

 6.1 Uniqueness and openness

 6.2 A priori estimates

 7 Symplectic Calabi-Yau surfaces

 7.1 Symplectic .birational geometry and Kodaira dimension

 7.2 Examples

 7.3 Homological classification

 7.4 Further questions

 References

Lectures on Stability and Constant Scalar Curvature

 D.H. Phong, Jacob Sturm

 1 Introduction

 2 The conjecture of Yau

 2.1 Constant scalar curvature metrics in a given Kahler class

 2.2 The special case of Kahler-Einstein metrics

 2.3 The conjecture of Yau

 3 The analytic problem

 3.1 Fourth order non-linear PDE and Monge-Amp~re

 equations

 3.2 Geometric heat flows

 3.3 Variational formulation and energy functionals

 4 The spaces/Ck of Bergman metrics

 4.1 Kodaira imbeddings

 4.2 The Tian-Yau-Zelditch theorem

 5 The functional F0w0 （φ） on Kk

 5.1 F0w0 （φ） and balance imbeddings

 5.2 F0w0 （φ） and the Euler-Lagrange equation R - R = 0

 5.3 F0w0 （φ） and Monge-Ampere masses

 6 Notions of stabilitY

 6.1 Stability in GIT

 6.2 Donaldson's infinite-dimensional GIT

 6.3 Stability conditions on Diff（X） orbits

 7 The necessity of stability

 7.1 The Moser-Trudinger inequality and analytic K-stability.

 7.2 Necessity of Chow-Mumford stability

 7.3 Necessity of semi K-stability

 8 Sufficient conditions： the Kahler-Einstein case

 8.1 The α-invariant

 8.2 Nadel's multiplier ideal sheaves criterion

 8.3 The Kahler-Ricci flow

 9 General L： energy functionals and Chow points

 9.1 F0w0 （φ） and Chow points

 9.2 Kw and Chow points

 10 General L： the Calabi energy and the Calabi flow

 10.1 The Calabi flow

 10.2 Extremal metrics and stability

 11 General L： toric varieties

 11.1 Symplectic potentials

 11.2 K-stability on toric varieties

 11.3 The K-unstable case

 12 Geodesics in the space K of Kahler potentials

 12.1 The Dirichlet problem for the complex Monge-Ampere

 equation

 12.2 Method of elliptic regularization and a priori estimates

 12.3 Geodesics in Kk and geodesics in Kk

 References

Analytic Aspect of Hamilton's Ricci Flow

 Xi-Ping Zhu

 Introduction

 1 Short-time existence and uniqueness

 2 Curvature estimates

 2.1 Shi's local derivative estimates

 2.2 Preserving positive curvature

 2.3 Hamilton-Ivey pinching estimate

 2.4 Li-Yau-Hamilton inequality

 3 Singularities of solutions

 3.1 Can all types of singularities be formed

 3.2 Singularity models

 3.3 Canonical neighborhood structure

 4 Long time behaviors

 4.1 The Ricci flow on two-manifolds

 4.2 The Ricci flow on three-manifolds

 4.3 Differential Sphere Theorems

 References