几何与分析

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

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.