几何与分析（第1卷） [Geometry and Analysis(Vol.I)] pdf epub mobi txt 电子书 下载 2025

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图书标签:

• 几何学
• 数学分析
• 微积分
• 拓扑学
• 实分析
• 函数论
• 高等数学
• 数学
• 几何
• 分析

出版社： 高等教育出版社

ISBN：9787040302721

版次：1

商品编码：10336140

包装：平装

外文名称：Geometry and Analysis(Vol.I)

开本：16开

出版时间：2010-09-01

用纸：胶版纸

页数：542

正文语种：英文

内容简介

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.

目录

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

精彩书摘

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.
……

好的，这是一份关于其他数学领域专著的详细简介，旨在避免提及《几何与分析（第1卷）》的内容，并以专业、详实的风格呈现。 --- 深入探索数学前沿：微分拓扑与黎曼几何的精妙结合 《流形上的几何结构与动力系统：理论构建与应用探析》 本书聚焦于现代微分几何与拓扑学的前沿交叉领域，系统阐述了流形上的度量结构、曲率理论，以及这些几何概念如何深刻地影响和揭示动力系统的长期行为。本书旨在为高年级本科生、研究生以及相关研究人员提供一个深入理解几何与分析如何相互作用的坚实基础。 第一部分：微分拓扑基础与光滑结构 本书的第一部分首先奠定坚实的拓扑与微分拓扑学基础。我们从一般拓扑空间的概念出发，逐步过渡到光滑流形的定义，详细讨论了切丛、余切丛以及向量丛的构造。重点章节深入探讨了微分形式的代数结构（如楔积），以及微分流形上的外微分运算（$d$算子）及其核心性质，特别是De Rham上同调理论。 本部分对浸入、淹没和横截性的讨论尤为详尽。通过对嵌入定理的严格证明，我们展示了光滑结构如何允许我们将抽象的拓扑空间映射到欧几里得空间中进行局部研究。此外，本书还详细考察了李群和李代数在几何中的作用，特别是作为保持流形结构（如等距变换群）的对称性工具。 关键概念深度剖析： 流形的分类与嵌入： 介绍Whitney嵌入定理和Smale的拓扑稳定性和可展性概念。 向量丛上的联络： 阐述切丛上的仿（仿射）联络的概念，定义黎曼联络的特殊性，以及曲率张量的产生机制。 第二部分：黎曼几何的核心理论 第二部分是本书的几何核心，完全致力于黎曼几何。我们从定义一个黎曼度量开始，精确地引入了Levi-Civita联络的唯一性及其构造。重点在于理解曲率的概念如何从欧几里得空间的平坦几何推广到弯曲空间。 测地线理论是本部分的核心主题之一。本书严格推导了测地线的变分性质，并利用这些性质引入指数映射。通过对指数映射的详细分析，我们建立了流形上局部坐标系和邻域结构之间的桥梁。此外，本书深入探讨了卡坦-阿达马德（Cartan-Hadamard）定理及其在零曲率和负曲率流形上的推论。 空间曲率的深化研究： 截面曲率与Ricci曲率： 对这些关键几何不变量进行了细致的计算和几何解释。特别关注Ricci曲率在线性化的引力理论和物质分布中的意义。 共边际、共形变换与Killing向量场： 详细分析了在保持度量结构不变的情况下允许的变换，特别关注Killing向量场在定义流形的对称性（等距运动）方面的关键作用。 极小曲面与极值原理： 引入了面积泛函，探讨了极小曲面作为该泛函的临界点，并通过Dirichlet能量等概念将分析工具引入到几何问题的研究中。 第三部分：几何与动力系统的交汇 本书的第三部分将前两部分建立的微分几何框架应用于动力系统的几何结构分析。我们不再将动力系统视为纯粹的常微分方程解的集合，而是将其视为作用在流形上的向量场。 拓扑动力学基础： 向量场与流： 定义流形上的向量场，并严格证明流的存在性和唯一性，引入庞加莱截面和李雅普诺夫指数的概念。 可积性与守恒量： 考察在黎曼度量下，向量场保持能量（Hamiltonian）的条件，探讨可积系统的几何特征，特别是Liouville可积性的结构。 曲率对混沌行为的影响： 本书的独特视角在于如何利用曲率信息来预测动力系统的长期稳定性或混沌性。 1. 负曲率与混沌（Chaos）： 详细分析了Pesin的熵公式在具有恒定负截面曲率的流形上的表现，揭示了负曲率如何自然地产生指数分离和混沌行为。 2. 杨-米尔斯理论的几何背景（选讲）： 简要介绍将黎曼几何扩展到纤维丛上的概念，侧重于规范联络和规范场方程的几何起源，这为理解粒子物理中的几何结构提供了分析框架。 3. 测地流的稳定性： 分析了在曲率不为零的流形上，测地流（Geodesic Flow）的性质。在正曲率下观察到轨道的收敛，而在负曲率下观察到轨道的剧烈分离，这是连接几何结构与动力学稳定性的直接桥梁。 总结与展望 《流形上的几何结构与动力系统》不仅是对经典几何理论的严谨回顾，更是对现代数学研究方向的一次有力展望。本书通过将拓扑的定性分析、黎曼几何的度量工具以及动力系统的演化视角相结合，提供了一个强大的多学科分析平台。读者将能够掌握如何利用曲率等内在几何量来精确描述和预测复杂系统的行为模式。本书的丰富例题和详细的证明过程，确保了其作为高级参考书的价值。 ---

评分☆☆☆☆☆

这本书的装帧和印刷质量绝对是业界良心。你很难想象，在如今这个追求快速迭代的时代，还能看到如此精良的纸张和装订工艺。打开书页，闻到那种淡淡的油墨香，立刻让人心神安定下来。作为一名对数学美学有执着追求的读者，我必须强调，书中对于那些复杂公式的排版处理得极其优雅。它们不是简单地堆砌在一起，而是被巧妙地布局在页面上，形成一种视觉上的平衡感，这本身就是一种艺术。我记得有一次，我在咖啡馆里阅读，旁边一位学物理的朋友凑过来看了一眼，他立刻表示赞叹，说这种严谨的排版是高等数学书籍的标配，但能做到如此赏心悦目的，实属罕见。这种对细节的极致追求，无形中也提升了阅读的专注度，让人更愿意长时间沉浸在这些数字和符号构建的世界里，去探寻那些隐藏在表象之下的深刻结构。

评分☆☆☆☆☆

这套书的封面设计简直是直击灵魂的复古感，那种厚重的纸张质感，拿在手里沉甸甸的，让人立刻感受到一股学术的庄重感。我不是数学专业的，但这套书的排版和字体选择，即便是初次接触，也透着一种莫名的亲切感。尤其是扉页上那几行引文，似乎在无声地诉说着数学的永恒之美。翻开第一页，那些精心绘制的几何图形，线条的流畅和角度的精准，真的让人叹为观止。我记得我当时为了理解其中一个关于拓扑空间的基础概念，花了整整一个下午，虽然过程有点烧脑，但最终豁然开朗的那一刻，那种知识被大脑吸收的喜悦，是任何通俗读物都无法比拟的。我特别喜欢它在介绍每一个定理时，都会附带一个简短的历史背景，仿佛在带领读者穿越时空，去感受那些伟大学者们是如何一步步构建起这座数学大厦的。这本书的厚度本身就是一种挑战，但当你沉浸其中，时间仿佛静止了，你所有的注意力都被那些严谨的逻辑和深邃的理论所吸引。

评分☆☆☆☆☆

说实话，当我第一次拿到《几何与分析（第1卷）》时，内心是有些抗拒的。毕竟“几何与分析”这几个字听起来就带着一股子高冷的精英气息，我担心自己无法驾入其深奥的门槛。然而，随着阅读的深入，我发现作者在构建理论体系时所展现出的耐心和细致，完全超出了我的预期。它不像某些教科书那样，直接扔出一堆公式让你死记硬背，而是像一位循循善诱的导师，每一步推导都讲解得清晰明了，逻辑链条环环相扣，几乎没有可以跳跃思考的空隙。我尤其欣赏它在引入新的数学工具时，总会先从一个直观的物理模型或一个简单的几何直觉出发，这极大地降低了抽象概念带来的学习障碍。我甚至发现，仅仅是阅读其中关于欧几里得空间中度量和范数的介绍部分，就让我对日常生活中遇到的“距离”和“大小”有了全新的、更深刻的理解。这本书不是用来快速阅读的，它是用来“消磨”时间的，用最好的方式。

评分☆☆☆☆☆

坦白说，我并非纯粹的数学研究者，我更多地关注它在跨学科应用上的潜力。这本书虽然基础扎实，但其内在的结构逻辑，对理解现代物理学的场论和信息论中的某些抽象模型有着极强的启发性。我曾尝试用书中的某个几何变换的概念去类比处理我工作中遇到的数据结构问题，结果发现那个模型简直是为我的困境量身定做的钥匙。这种在看似不相关的领域间搭建桥梁的能力，正是顶尖数学著作的魅力所在。它教会你辨识不同领域中的同构性，理解万物底层逻辑的统一性。这本书的内容深度，足以让一个研究生课题有所启发，但其表达方式的清晰度，又保证了本科高年级学生能够跟进。它像一座灯塔，为在知识海洋中摸索的人提供了一个稳定且光芒四射的参照点，指引我们前进的方向。

评分☆☆☆☆☆

这本书给我的感觉，更像是一部数学思想的编年史，而非单纯的公式汇编。它不仅仅是在教你“如何算”，更是在教你“如何思考”。我特别留意到作者在论述微积分基础概念时，引用了费马和牛顿等前辈的观点，对比了不同历史时期对“极限”这个核心概念的认识演变。这种跨越时代的对话，让原本冰冷的数学概念立刻鲜活了起来，充满了人性的探索欲和智慧的火花。我过去总觉得分析学是枯燥的分析，但阅读此书后，我开始理解为什么人们称之为“分析”，因为它要求你将事物拆解到最细微的颗粒度，然后用最严谨的逻辑重新审视和组合。对于那些希望真正掌握数学语言精髓的人来说，这本书提供的不仅仅是知识，更是一种思维模式的重塑，它训练你的批判性思维，让你不满足于表面的结论，而要追溯到最根本的公理。