幾何與分析（第1捲） [Geometry and Analysis(Vol.I)] pdf epub mobi txt 電子書 下載 2025

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☆☆☆☆☆

季理真 編

圖書標籤:

• 幾何學
• 數學分析
• 微積分
• 拓撲學
• 實分析
• 函數論
• 高等數學
• 數學
• 幾何
• 分析

齣版社： 高等教育齣版社

ISBN：9787040302721

版次：1

商品編碼：10336140

包裝：平裝

外文名稱：Geometry and Analysis(Vol.I)

開本：16開

齣版時間：2010-09-01

用紙：膠版紙

頁數：542

正文語種：英文

內容簡介

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

精彩書摘

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，這是一份關於其他數學領域專著的詳細簡介，旨在避免提及《幾何與分析（第1捲）》的內容，並以專業、詳實的風格呈現。 --- 深入探索數學前沿：微分拓撲與黎曼幾何的精妙結閤 《流形上的幾何結構與動力係統：理論構建與應用探析》 本書聚焦於現代微分幾何與拓撲學的前沿交叉領域，係統闡述瞭流形上的度量結構、麯率理論，以及這些幾何概念如何深刻地影響和揭示動力係統的長期行為。本書旨在為高年級本科生、研究生以及相關研究人員提供一個深入理解幾何與分析如何相互作用的堅實基礎。 第一部分：微分拓撲基礎與光滑結構 本書的第一部分首先奠定堅實的拓撲與微分拓撲學基礎。我們從一般拓撲空間的概念齣發，逐步過渡到光滑流形的定義，詳細討論瞭切叢、餘切叢以及嚮量叢的構造。重點章節深入探討瞭微分形式的代數結構（如楔積），以及微分流形上的外微分運算（$d$算子）及其核心性質，特彆是De Rham上同調理論。 本部分對浸入、淹沒和橫截性的討論尤為詳盡。通過對嵌入定理的嚴格證明，我們展示瞭光滑結構如何允許我們將抽象的拓撲空間映射到歐幾裏得空間中進行局部研究。此外，本書還詳細考察瞭李群和李代數在幾何中的作用，特彆是作為保持流形結構（如等距變換群）的對稱性工具。 關鍵概念深度剖析： 流形的分類與嵌入： 介紹Whitney嵌入定理和Smale的拓撲穩定性和可展性概念。 嚮量叢上的聯絡： 闡述切叢上的仿（仿射）聯絡的概念，定義黎曼聯絡的特殊性，以及麯率張量的産生機製。 第二部分：黎曼幾何的核心理論 第二部分是本書的幾何核心，完全緻力於黎曼幾何。我們從定義一個黎曼度量開始，精確地引入瞭Levi-Civita聯絡的唯一性及其構造。重點在於理解麯率的概念如何從歐幾裏得空間的平坦幾何推廣到彎麯空間。 測地綫理論是本部分的核心主題之一。本書嚴格推導瞭測地綫的變分性質，並利用這些性質引入指數映射。通過對指數映射的詳細分析，我們建立瞭流形上局部坐標係和鄰域結構之間的橋梁。此外，本書深入探討瞭卡坦-阿達馬德（Cartan-Hadamard）定理及其在零麯率和負麯率流形上的推論。 空間麯率的深化研究： 截麵麯率與Ricci麯率： 對這些關鍵幾何不變量進行瞭細緻的計算和幾何解釋。特彆關注Ricci麯率在綫性化的引力理論和物質分布中的意義。 共邊際、共形變換與Killing嚮量場： 詳細分析瞭在保持度量結構不變的情況下允許的變換，特彆關注Killing嚮量場在定義流形的對稱性（等距運動）方麵的關鍵作用。 極小麯麵與極值原理： 引入瞭麵積泛函，探討瞭極小麯麵作為該泛函的臨界點，並通過Dirichlet能量等概念將分析工具引入到幾何問題的研究中。 第三部分：幾何與動力係統的交匯 本書的第三部分將前兩部分建立的微分幾何框架應用於動力係統的幾何結構分析。我們不再將動力係統視為純粹的常微分方程解的集閤，而是將其視為作用在流形上的嚮量場。 拓撲動力學基礎： 嚮量場與流： 定義流形上的嚮量場，並嚴格證明流的存在性和唯一性，引入龐加萊截麵和李雅普諾夫指數的概念。 可積性與守恒量： 考察在黎曼度量下，嚮量場保持能量（Hamiltonian）的條件，探討可積係統的幾何特徵，特彆是Liouville可積性的結構。 麯率對混沌行為的影響： 本書的獨特視角在於如何利用麯率信息來預測動力係統的長期穩定性或混沌性。 1. 負麯率與混沌（Chaos）： 詳細分析瞭Pesin的熵公式在具有恒定負截麵麯率的流形上的錶現，揭示瞭負麯率如何自然地産生指數分離和混沌行為。 2. 楊-米爾斯理論的幾何背景（選講）： 簡要介紹將黎曼幾何擴展到縴維叢上的概念，側重於規範聯絡和規範場方程的幾何起源，這為理解粒子物理中的幾何結構提供瞭分析框架。 3. 測地流的穩定性： 分析瞭在麯率不為零的流形上，測地流（Geodesic Flow）的性質。在正麯率下觀察到軌道的收斂，而在負麯率下觀察到軌道的劇烈分離，這是連接幾何結構與動力學穩定性的直接橋梁。 總結與展望 《流形上的幾何結構與動力係統》不僅是對經典幾何理論的嚴謹迴顧，更是對現代數學研究方嚮的一次有力展望。本書通過將拓撲的定性分析、黎曼幾何的度量工具以及動力係統的演化視角相結閤，提供瞭一個強大的多學科分析平颱。讀者將能夠掌握如何利用麯率等內在幾何量來精確描述和預測復雜係統的行為模式。本書的豐富例題和詳細的證明過程，確保瞭其作為高級參考書的價值。 ---

評分☆☆☆☆☆

這本書給我的感覺，更像是一部數學思想的編年史，而非單純的公式匯編。它不僅僅是在教你“如何算”，更是在教你“如何思考”。我特彆留意到作者在論述微積分基礎概念時，引用瞭費馬和牛頓等前輩的觀點，對比瞭不同曆史時期對“極限”這個核心概念的認識演變。這種跨越時代的對話，讓原本冰冷的數學概念立刻鮮活瞭起來，充滿瞭人性的探索欲和智慧的火花。我過去總覺得分析學是枯燥的分析，但閱讀此書後，我開始理解為什麼人們稱之為“分析”，因為它要求你將事物拆解到最細微的顆粒度，然後用最嚴謹的邏輯重新審視和組閤。對於那些希望真正掌握數學語言精髓的人來說，這本書提供的不僅僅是知識，更是一種思維模式的重塑，它訓練你的批判性思維，讓你不滿足於錶麵的結論，而要追溯到最根本的公理。

評分☆☆☆☆☆

這套書的封麵設計簡直是直擊靈魂的復古感，那種厚重的紙張質感，拿在手裏沉甸甸的，讓人立刻感受到一股學術的莊重感。我不是數學專業的，但這套書的排版和字體選擇，即便是初次接觸，也透著一種莫名的親切感。尤其是扉頁上那幾行引文，似乎在無聲地訴說著數學的永恒之美。翻開第一頁，那些精心繪製的幾何圖形，綫條的流暢和角度的精準，真的讓人嘆為觀止。我記得我當時為瞭理解其中一個關於拓撲空間的基礎概念，花瞭整整一個下午，雖然過程有點燒腦，但最終豁然開朗的那一刻，那種知識被大腦吸收的喜悅，是任何通俗讀物都無法比擬的。我特彆喜歡它在介紹每一個定理時，都會附帶一個簡短的曆史背景，仿佛在帶領讀者穿越時空，去感受那些偉大學者們是如何一步步構建起這座數學大廈的。這本書的厚度本身就是一種挑戰，但當你沉浸其中，時間仿佛靜止瞭，你所有的注意力都被那些嚴謹的邏輯和深邃的理論所吸引。

評分☆☆☆☆☆

坦白說，我並非純粹的數學研究者，我更多地關注它在跨學科應用上的潛力。這本書雖然基礎紮實，但其內在的結構邏輯，對理解現代物理學的場論和信息論中的某些抽象模型有著極強的啓發性。我曾嘗試用書中的某個幾何變換的概念去類比處理我工作中遇到的數據結構問題，結果發現那個模型簡直是為我的睏境量身定做的鑰匙。這種在看似不相關的領域間搭建橋梁的能力，正是頂尖數學著作的魅力所在。它教會你辨識不同領域中的同構性，理解萬物底層邏輯的統一性。這本書的內容深度，足以讓一個研究生課題有所啓發，但其錶達方式的清晰度，又保證瞭本科高年級學生能夠跟進。它像一座燈塔，為在知識海洋中摸索的人提供瞭一個穩定且光芒四射的參照點，指引我們前進的方嚮。

評分☆☆☆☆☆

說實話，當我第一次拿到《幾何與分析（第1捲）》時，內心是有些抗拒的。畢竟“幾何與分析”這幾個字聽起來就帶著一股子高冷的精英氣息，我擔心自己無法駕入其深奧的門檻。然而，隨著閱讀的深入，我發現作者在構建理論體係時所展現齣的耐心和細緻，完全超齣瞭我的預期。它不像某些教科書那樣，直接扔齣一堆公式讓你死記硬背，而是像一位循循善誘的導師，每一步推導都講解得清晰明瞭，邏輯鏈條環環相扣，幾乎沒有可以跳躍思考的空隙。我尤其欣賞它在引入新的數學工具時，總會先從一個直觀的物理模型或一個簡單的幾何直覺齣發，這極大地降低瞭抽象概念帶來的學習障礙。我甚至發現，僅僅是閱讀其中關於歐幾裏得空間中度量和範數的介紹部分，就讓我對日常生活中遇到的“距離”和“大小”有瞭全新的、更深刻的理解。這本書不是用來快速閱讀的，它是用來“消磨”時間的，用最好的方式。

評分☆☆☆☆☆

這本書的裝幀和印刷質量絕對是業界良心。你很難想象，在如今這個追求快速迭代的時代，還能看到如此精良的紙張和裝訂工藝。打開書頁，聞到那種淡淡的油墨香，立刻讓人心神安定下來。作為一名對數學美學有執著追求的讀者，我必須強調，書中對於那些復雜公式的排版處理得極其優雅。它們不是簡單地堆砌在一起，而是被巧妙地布局在頁麵上，形成一種視覺上的平衡感，這本身就是一種藝術。我記得有一次，我在咖啡館裏閱讀，旁邊一位學物理的朋友湊過來看瞭一眼，他立刻錶示贊嘆，說這種嚴謹的排版是高等數學書籍的標配，但能做到如此賞心悅目的，實屬罕見。這種對細節的極緻追求，無形中也提升瞭閱讀的專注度，讓人更願意長時間沉浸在這些數字和符號構建的世界裏，去探尋那些隱藏在錶象之下的深刻結構。