几何分析手册. 第3卷

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 ModuⅡ 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 Hamiltons 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 equations

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 Kaihler 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 submanifolds 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

10.1 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 WeitzenbSck 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 Furutas finite dim. Approximations

5.6 SW invariants

5.7 Symplectic SW equations 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 Homologieal classification

7.4 Further questions

References

Lectures on Stability and Constant Scalar Curvature

D.H. Phong， Jacob $turm

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-Ampere equations

3.2 Geometric heat flows

3.3 Variational formulation and energy functionals

4 The spaces Kk of Bergman metrics

4.1 Kodaira imbeddings

4.2 The Tian-Yau-Zelditch theorem

5 The functional F0ω0 on Kk

5.1 F0ω0 and balance imbeddings

5.2 F0ω0 and the Euler-Lagrange equation R-R = 0

5.3 F0ω0 and Monge-Ampere masses

6 Notions of stability

6.1 Stability in GIT

6.2 Donaldsons 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 KⅡhler-Einstein case

8.1 The α-invariant

8.2 Nadels multiplier ideal sheaves criterion

8.3 The Kahler-Ricci flow

9 General L： energy functionals and Chow points

9.1 F0ω 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/g of Kaihler 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/g and geodesics in/gk

References

Analytic Aspect of Hamiltons Ricci Flow

Xi-Ping Zhu

Introduction

1 Short-time existence and uniqueness

2 Curvature estimates

2.1 Shis 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

Geometric Analysis combines differential equations and differential geometry. An important 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. A particularly important example is the Monge-Ampere equation. Applications to geometric problems have also motivated new methods and techniques in differential equations. The field of geometric analysis is broad and has had many striking applications. This handbook of geometric analysis provides introductions to and surveys of important topics in geometric analysis and their applications to related fields which is intend to be referred by graduate students and researchers in related areas.

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