幾何與分析（第I卷）

內(nèi)容概要

　　This book contains many substantial papers from distinguished speakers of a conference "Geometric Analysis: Present and Future" and an overview of the works of Professor Shing-Tung Yau. Contributors include E. Wit-ten, Y.T. Siu, R. Hamilton, H. Hitchin, B. Lawson, A. Strominger, C. Vafa, W. Schmid, V. Guillemin, N. Mok, D. Christodoulou. This is a valuable reference that gives an up-to-dated summary of geometric analysis and its applications in many different areas of mathematics.

書籍目錄

part 1 summary of and commentaries on the work of shing-tung yau curriculum vitae of shing-tung yau a brief overview of the work of shing-tung yau lizhen ji 1 introduction 2 a summary of some major works of yau 3 topics yau has worked on 4 basics on kaihler-einstein metrics and calabi conjectures 5 some applications of kaihler-einstein metrics and calabi-yau manifolds 6 harmonic maps 7 rigidity of kahler manifolds 8 super-rigidity of spaces of nonpositive curvature 9 survey papers by yau 10 open problems by yau ll books written and co-written by yau 12 books edited and co-edited by yau 13 ph.d. students of yau 14 partial list of papers and books of yau references yau's work on filtering problem .wen-lin chiou, jie huang and lizhen ji 1 filtering problem 2 yau's two methods in solving nonlinear filtering problem 2.1 direct method 2.2 algorithm for real time solution without memory references from continues to discrete - yau's work on graph theory fan chung yau's work on moduli, periods, and mirror maps for calabi-yau manifolds charles f. doran 1 construction of calabi-yau threefolds 2 picard-fuchs equations and the mirror map 3 arithmetic properties of mirror maps 4 periods and moduli of complex tori and k3 surfaces references review on yau's work on the coupled einstein equations and the wave dynamics in the kerr geometry felix finster 1 coupling the einstein equations to non-abelian gauge fields and dirac spinors 2 the dynamics of linear waves in the kerr geometry references the work of witten and yau on the ads/cft correspondence gregory j. galloway 1 introduction 2 the witten-yau results on ads/cft 3 further developments references yau's work on heat kernels alexander grigor'yan 1 the notion of the heat kernel 2 estimating heat kernels 3 some applications of the heat kernel estimates references yau's contributions to engineering fields xianfeng david gu 1 introduction 2 computational conformal geometry 2.1 conformal structure 2.2 harmonic map 2.3 surfacericci flow 2.4 conformal mappings 2.5 quasi-conformal mappings 2.6 teichmiiller space 3 geometric acquisition 4 computer graphics 5 geometric modeling 6 medical imaging 7 computer vision 8 wireless sensor network 9 summary references the syz proposal naichung conan leung 1 pre-syz 2 the birth of syz 3 the growing up of syz 3.1 special lagrangian geometry 3.2 special lagrangian fibrations 3.3 affine geometry 3.4 syz transformation 4 future of syz references yau- zaslow formula naichung conan leung yau's work on function theory: harmonic functions, eigenvalues and the heat equation peter li a vision of yau on mirror symmetry bong lian 1 enumerative geometry 2 geometry of calabi-yau manifolds and their moduli spaces references yau's work on group actions kefeng liu cheng and yau's work on the monge-ampere equation and affine geometry john loftin, xu-jia wang and deane yang 1 introduction 2 the monge-ampere equation 3 cheng and yau's work on the dirichlet problem 4 subsequent work on the monge-ampere equation 5 affine spheres 6 hyperbolic affine spheres and real monge-ampere equations 7 affine manifolds 8 maximal hypersurfaces in minkowski space 9 the minkowski problem 10 convex geometry without smoothness assumptions 10.1 support function 10.2 invariance properties of the support function 10.3 minkowski sum 10.4 mixed volume 10.5 surface area measure 10.6 invariance properties of the surface area measure 10.7 the minkowski problem 10.8 the brunn-minkowski inequality 10.9 uniqueness in the minkowski problem 10.10 variational approach to the minkowski problem 11 convex geometry with smoothness assumptions 11.1 the inverse gauss map 11.2 the inverse second fundamental form 11.3 the curvature function 11.4 the surface area measure 11.5 the minkowski problem 11.6 the minkowski problem as a pde 12 cheng and yau's regularity theorem for the minkowski problem. 12.1 statement 12.2 sketch of proof 13 generalizations of the minkowski problem references yau's work on minimal surfaces and 3-manifolds feng luo the work of schoen and yau on manifolds with positive scalar curvature william, minicozzi ii 0 introduction 1 topological restrictions on manifolds with positive scalar curvature 1.1 stable minimal surfaces and scalar curvature 1.2 inductively extending this to higher dimensions 1.3 preserving positive scalar curvature under surgery 2 locally conformally flat manifolds 2.1 the new invariants 2.2 a positive mass theorem references yau's contributions to algebraic geometry ndrey todorov 1 introduction 1.1 yau's program-plenary talk at icm 1982 2 monge-ampere equation and applications to algebraic geometry 2.1 solution of the calabi conjecture 2.2 existence of canonical metrics on zariski open sets 3 stable vector bundles over kahler manifolds 3.1 donaldson-uhlenbeek-yau theorem 3.2 applications to kodaira's classification of surfaces 4 moduli spaces 4.1 existence of kiihler-einstein metrics on domain of holomorphy and teichmfiller spaces 4.2 moduli spaces of k3 surfaces 4.3 moduli spaces of cy manifolds 4.4 generalization of shwarz lemma by yau and baily-borel compactification 5 contributions of yau to string theory 5.1 mirror symmetry and syz conjecture 5.2 large radius limit 5.3 string theory and number theory 5.4 rational curves on algebraic k3 surfaces 6 rigidity 6.1 yau's conjecture about rigidity of some complex manifolds 6.2 geometric proof of margulis' superrigidity 6.3 geometric proof of kazhdan theorem about galois conjugation of shimura varieties references yau's work on positive mass theorems mu-tao wang yan's conjecture on kaihler-einstein metric and stability xiaowei wang on yau's pioneer contribution on the frankel conjecture and related questions fangyang zheng yau's work on inequalities between chern numbers and uniformization of complex manifolds kang zuo part 2 differential geometry and differential equations geometry of complete gradient shrinking ricci solitons huai-dong cao 1 gradient shrinking ricci solitons 2 classification of 3-dimensional gradient shrinking solitons 3 geometry of complete gradient solitons references the formation of black holes in general relativity demetrios christodoulou pagerank as a discrete green's function fan chung 1 introduction 2 preliminaries 3 dirichlet eigenvalues 4 connections between pagerank and discrete green's function 5 relating the cheeger constant to the pagerank 6 relating the pagerank of a graph to that of its subgraphs 7 the pagerank and the hitting time references a geodesic equation in the space of sasakian metrics pengfei guan and xi zhang some inverse spectral results for the two-dimensional schrodinger operator v. cuillemin and a. uribe 1 introduction 2 the weyl calculus 3 some bracket identities 4 the quantum birkhoff canonical form references li-yau estimates and their harnack inequalities richard s. hamilton 1 the heat equation 2 the dirichlet problem for the heat equation 3 the heat equation in the plane 4 the castaway 5 endangered species equation 6 the migration equation 7 motion of a curve by its curvature 8 motion of a surface by its mean curvature 9 motion of a surface by its gauss curvature references plurisubharmonicity in a general geometric context f. reese harvey and h. blaine lawson, jr 1 introduction 2 geometrically defined plurisubharmonic functions 3 more general plurisubharmonic functions defined by an elliptic cone p+ 4 p+-plurisubharmonic distributions 5 upper-semi-continuous p+-plurisubharmonic functions 6 some classical facts that extend to p+-plurisubharmonie functions 7 the dirichlet problem uniqueness 8 the dirichlet problem existence 9 p+-convex domains 10 topological restrictions on p+-convex domains 11 p+-free submanifolds 12 p+-convex boundaries references poisson modules and generalized geometry nigel hitchin 1 introduction 2 poisson modules 2.1 definitions 2.2 a construction 3 the serre construction 3.1 the algebraic approach 3.2 the analytical approach 3.3 the second section 4 generalized geometry 4.1 basic features 4.2 generalized dolbeault operators 4.3 the canonical bundle 5 a generalized construction 5.1 the problem 5.2 generalized complex submanifolds 5.3 the construction 6 an application references uniqueness of solutions to mean field equations of liouville type in two-dimension chang-shou lin 1 introduction 2 uniqueness in r2 3 uniqueness in bounded domains of r2 4 onofri inequality and its generalization 5 mean field equation and green functions on torus 6 generalized liouville system references monotonicity and holomorphic functions lei ni decay of solutions to the cauchy problem in the kerr geometry for various physical systems: stability of black holes j. a. smoller 1 introduction 2 main references the calabi-yau equation, symplectic forms and almost complex structures valentino tosatti and ben weinkove 1 background- yau's theorem 2 donaldson's conjecture and applications 3 estimates for the catabi-yau equation 4 methods 5 a monotonicity formula references understanding weil-petersson curvature scott a. wolpert 1 introduction 2 basics of teichmiiller theory 3 wp intrinsic geometry 4 methods 5 applications of curvature 5.1 the work of liu, sun and yau 5.2 the model metric 4dr2 + rs do2 5.3 projection and distance to a stratum references examples of positively curved complete kahler manifolds hung-hsi wu and fangyang zheng 1 introduction 2 the abcd functions 3 characterization by the function 4 some examples 5 characterization by surface of revolution 6 correlation between volume growth and curvature decay references

章節(jié)摘錄

插圖：hough geometric analysis has a long history, the decisive contributions of Yau since 1970s have made it an indispensable tool in many subjects such as differential geometry, topology, algebraic geometry, mathematical physics, etc, and hence have established it as one of the most important fields of modern mathematics.Yau's impacts are clearly visible in the papers of these two volumes, and we hope that these two volumes of Geometry and Analysis and the three volumes of the Handbook of Geometric Analysis will pay a proper tribute to him in a modest way. According to the Chinese tradition, a person is one year old when he is born, and hence Yau turned 60 already in 2008. The number 60 and hence the age 60 is special in many cultures, especially in the Chinese culture. It is the smallest common multiple of 10 and 12, two important periods in the Chinese astronomy. Therefore, it is a new starting point (or a new cycle). A quick look at Yau's list of publications in Part 1 shows that Yau has not only maintained but increased his incredible output both in terms of quality and quantity.

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