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2020-01-21 14:43 來源：京東作者：京東

編輯推薦: 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.

內(nèi)容簡介: 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.

作者簡介:

目錄: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

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