几何分析手册
2010-2
高等教育出版社
季理真 编
472
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The marriage of geometry and analysis， in particular non-linear differentialequations， has been very fruitful. An early deep application of geometric analysisis the celebrated solution by Shing-Tung Yau of the Calabi conjecture in 1976. Infact， Yau together with many of his collaborators developed important techniquesin geometric analysis in order to solve the Calabi conjecture. Besides solvingmany open problems in algebraic geometry such as the Severi conjecture， thecharacterization of complex projective varieties， and characterization of certainShimura varieties， the Calabi-Yau manifolds also provide the basic building blocksin the superstring theory model bf the universe. Geometric analysis has alsobeen crucial in solving many outstanding problems in low dimensional topology，for example， the Smith conjecture， and the positive mass conjecture in generalrelativity. Geometric analysis has been intensively studied and highly developed since1970s， and it is becoming an indispensable tool for understanding many partsof mathematics. Its success also brings with it the difficulty for the uninitiatedto appreciate its breadth and depth. In order to introduce both beginners andnon-experts to this fascinating subject， we have decided to edit this handbook ofgeometric analysis. Each article is written by a leading expert in the field and willserve as both an introduction to and a survey of the topics under discussion. Thehandbook of geometric analysis is divided into several parts， and this volume isthe third part. Shing-Tung Yau has been crucial to many stages of the development of geo-metric analysis. Indeed， his work has played an important role in bringing thewell-deserved global recognition by the whole mathematicM sciences communityto the field of geometric analysis. In view of this， we would like to dedicate thishandbook of geometric analysis to Shing-Tung Yau on the occasion of his sixtiethbirthday.
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.
A Survey of Einstein Metrics on 4-manifoldsMichael T. Anderson1 Introduction2 Brief review： 4-manifolds， complex surfaces and Einstein metrics3 Constructions of Einstein metrics I4 Obstructions to Einstein metrics5 Moduli spaces I6 ModuⅡ spaces Ⅱ7 Constructions of Einstein metrics Ⅱ8 Concluding remarksReferencesSphere Theorems in GeometrySimon Brendle， Richard Schoen1 The Topological Sphere Theorem2 Manifolds with positive isotropic curvature3 The Differentiable Sphere Theorem4 New invariant curvature conditions for the Ricci flow5 Rigidity results and the classification of weakly 1/4-pinched manifolds6 Hamiltons differential Harnack inequality for the Ricci flow7 Compactness of pointwise pinched manifoldsReferencesCurvature Flows and CMC HypersurfacesClaus Gerhardt1 Introduction2 Notations and preliminary results3 Evolution equations for some geometric quantities.4 Essential parabolic flow equations5 Existence results6 Curvature flows in Riemannian manifolds7 Foliation of a spacetime by CMC hypersurfaces8 The inverse mean curvature flow in Lorentzian spacesReferencesGeometric Structures on Riemannian ManifoldsNaichung Conan Leung1 Introduction2 Topology of manifolds2.1 Cohomology and geometry of differential forms2.2 Hodge theorem2.3 Witten-Morse theory2.4 Vector bundles and gauge theory3 Riemannian geometry3.1 Torsion and Levi-Civita connections3.2 Classification of Riemannian holonomy groups3.3 Riemannian curvature tensors3.4 Flat tori3.5 Einstein metrics3.6 Minimal submanifolds3.7 Harmonic maps4 Oriented four manifolds4.1 Gauge theory in dimension four4.2 Riemannian geometry in dimension four5 Kaihler geometry5.1 Kahler geometry —— complex aspects5.2 Kahler geometry —— Riemannian aspects5.3 Kahler geometry —— symplectic aspects5.4 Gromov-Witten theory6 Calabi-Yau geometry6.1 Calabi-Yau manifolds6.2 Special Lagrangian geometry6.3 Mirror symmetry6.4 K3 surfaces7 Calabi-Yau 3-folds7.1 Moduli of CY threefolds7.2 Curves and surfaces in Calabi-Yau threefolds7.3 Donaldson-Thomas bundles over Calabi-Yau threefolds.7.4 Special Lagrangian submanifolds in CY37.5 Mirror symmetry for Calabi-Yau threefolds8 G2-geometry8.1 G2-manifolds8.2 Moduli of G2-manifolds8.3 （Co-）associative geometry8.4 G2-Donaldson-Thomas bundles8.5 G2-dualities， trialities and M-theory9 Geometry of vector cross products9.1 VCP manifolds9.2 Instantons and branes9.3 Symplectic geometry on higher dimensional knot spaces.9.4 C-VCP geometry9.5 Hyperkahler geometry on isotropic knot spaces of CY10 Geometry over normed division algebras10.1 Manifolds over normed algebras10.2 Gauge theory over （special） A-manifolds10.3 A-submanifolds and （special） Lagrangian submanifolds.11 Quaternion geometry11.1 Hyperkahler geometry11.2 Quaternionic-Kahler geometry12 Conformal geometry13 Geometry of Riemannian symmetric spaces13.1 Riemannian symmetric spaces13.2 Jordan algebras and magic square13.3 Hermitian and quaternionic symmetric spaces14 ConclusionsReferencesSymplectic Calabi-Yau SurfacesTian-Jun Li1 Introduction2 Linear symplectic geometry2.1 Symplectic vector spaces2.2 Compatible complex structures2.3 Hermitian vector spaces2.4 4-dimensional geometry3 Symplectic manifolds3.1 Almost symplectic and almost complex structures3.2 Cohomological invariants and space of symplectic structures3.3 Moser stability and Darboux charts3.4 Submanifolds and their neighborhoods3.5 Constructions4 Almost Kahler geometry4.1 Almost Hermitian manifolds， integrability and operators.4.2 Levi-Civita connection4.3 Connections and curvature on Hermitian bundles4.4 Chern connection and Hermitian curvatures4.5 The self-dual operator4.6 Dirac operators4.7 WeitzenbSck formulas and some almost Kahler identities.5 Seiberg-Witten theory-three facets5.1 SW equations5.2 Pin（2） symmetry for a spin reduction5.3 The compactness and Hausdorff property of the moduli space5.4 Generic smoothness of the moduli space5.5 Furutas finite dim. Approximations5.6 SW invariants5.7 Symplectic SW equations and Taubes nonvanishing5.8 Symplectic SW solutions and Pseudo-holomorphic curves.5.9 Bordism SW invariants via finite dim. Approximations5.10 Mod 2 vanishing and homology type6 Symplectic Calabi-Yau equation6.1 Uniqueness and openness6.2 A priori estimates7 Symplectic Calabi-Yau surfaces7.1 Symplectic birational geometry and Kodaira dimension7.2 Examples7.3 Homologieal classification7.4 Further questionsReferencesLectures on Stability and Constant Scalar CurvatureD.H. Phong， Jacob $turm1 Introduction2 The conjecture of Yau2.1 Constant scalar curvature metrics in a given Kahler class.2.2 The special case of Kahler-Einstein metrics2.3 The conjecture of Yau3 The analytic problem3.1 Fourth order non-linear PDE and Monge-Ampere equations3.2 Geometric heat flows3.3 Variational formulation and energy functionals4 The spaces Kk of Bergman metrics4.1 Kodaira imbeddings4.2 The Tian-Yau-Zelditch theorem5 The functional F0ω0 on Kk5.1 F0ω0 and balance imbeddings5.2 F0ω0 and the Euler-Lagrange equation R-R = 05.3 F0ω0 and Monge-Ampere masses6 Notions of stability6.1 Stability in GIT6.2 Donaldsons infinite-dimensional GIT6.3 Stability conditions on Diff（X） orbits7 The necessity of stability7.1 The Moser-Trudinger inequality and analytic K-stability7.2 Necessity of Chow-Mumford stability7.3 Necessity of semi K-stability8 Sufficient conditions： the KⅡhler-Einstein case8.1 The α-invariant8.2 Nadels multiplier ideal sheaves criterion8.3 The Kahler-Ricci flow9 General L： energy functionals and Chow points9.1 F0ω and Chow points9.2 Kw and Chow points10 General L： the Calabi energy and the Calabi flow10.1 The Calabi flow10.2 Extremal metrics and stability11 General L： toric varieties11.1 Symplectic potentials11.2 K-stability on toric varieties11.3 The K-unstable case12 Geodesics in the space/g of Kaihler potentials12.1 The Dirichlet problem for the complex Monge-Ampere equation12.2 Method of elliptic regularization and a priori estimates12.3 Geodesics in/g and geodesics in/gkReferencesAnalytic Aspect of Hamiltons Ricci FlowXi-Ping ZhuIntroduction1 Short-time existence and uniqueness2 Curvature estimates2.1 Shis local derivative estimates2.2 Preserving positive curvature2.3 Hamilton-Ivey pinching estimate2.4 Li-Yau-Hamilton inequality3 Singularities of solutions3.1 Can all types of singularities be formed3.2 Singularity models3.3 Canonical neighborhood structure4 Long time behaviors4.1 The Ricci flow on two-manifolds4.2 The Ricci flow on three-manifolds4.3 Differential Sphere TheoremsReferences
The launch of this Advanced Lectures in Mathematics series is aimed at keepingmathematicians informed of the latest developments in mathematics， as well asto aid in the learning of new mathematical topics by students all over the world.Each volume consists of either an expository monograph or a collection of signifi-cant introductions to important topics. This series emphasizes the history andsources of motivation for the topics under discussion， and also gives an overviewof the current status of research in each particular field. These volumes are thefirst source to which people will turn in order to learn new subjects and to dis-cover the latest results of many cutting-edge fields in mathematics.
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