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El. knyga: Differential Geometry in the Large

Edited by , Edited by (University of Adelaide), Edited by (Karlsruhe Institute of Technology, Germany), Edited by (University of Melbourne), Edited by
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The 2019 'Australian-German Workshop on Differential Geometry in the Large' represented an extraordinary cross section of topics across differential geometry, geometric analysis and differential topology. The two-week programme featured talks from prominent keynote speakers from across the globe, treating geometric evolution equations, structures on manifolds, non-negative curvature and Alexandrov geometry, and topics in differential topology. A joy to the expert and novice alike, this proceedings volume touches on topics as diverse as Ricci and mean curvature flow, geometric invariant theory, Alexandrov spaces, almost formality, prescribed Ricci curvature, and Kähler and Sasaki geometry.

Recenzijos

'The high-quality surveys and original work in this book give a convenient path to understand some recent exciting developments in global Differential Geometry and Geometric Analysis. This should be of great value to graduate students entering the field, as well as to more experienced researchers looking for an updated perspective on a wide range of topics, ranging from nonnegative curvature and Alexandrov spaces to geometric flows and equivariant geometry.' Renato G. Bettiol, Lehman College, The City University of New York 'The volume includes important additions to the literature including new results, new proofs of previous results, and simplified expositions, and also an excellent collection of surveys on recent activity. It is well written and offers a generous overview and invitation to a variety of modern, active topics in differential geometry.' Christopher Seaton, MAA Reviews

Daugiau informacijos

From Ricci flow to GIT, physics to curvature bounds, Sasaki geometry to almost formality. This is differential geometry at large.
List of Contributors
xv
Introduction 1(8)
Part One Geometric Evolution Equations and Curvature Flow
9(132)
1 Real Geometric Invariant Theory
11(39)
C. Bohm
R.A. Lafuente
1.1 Introduction
11(5)
1.2 Examples
16(2)
1.3 Comparison with Complex and Symplectic Case
18(1)
1.4 The Abelian Case
19(4)
1.5 Separation of Closed T-Invariant Sets
23(2)
1.6 The General Case of Real Reductive Groups
25(3)
1.7 Stratification
28(8)
1.8 Properties of Critical Points of the Energy Map
36(3)
1.9 Applications
39(2)
1.10 Appendices
41(9)
1.10.1 Real Reductive Lie Groups
41(3)
1.10.2 The Parabolic Subgroup
44(3)
References
47(3)
2 Convex Ancient Solutions to Mean Curvature Flow
50(25)
T. Bourni
M. Longford
G. Tinaglia
2.1 Introduction
50(2)
2.2 Asymptotics for Convex Ancient Solutions
52(4)
2.3 X.-J. Wang's Dichotomy for Convex Ancient Solutions
56(10)
2.4 Convex Ancient Solutions to Curve Shortening Flow
66(1)
2.5 Rigidity of the Shrinking Sphere
67(1)
2.6 Asymptotics for Convex Translators
68(2)
2.7 X.-J. Wang's Dichotomy for Convex Translators
70(1)
2.8 Rigidity of the Bowl Soliton
71(4)
References
72(3)
3 Negatively Curved Three-Manifolds, Hyperbolic Metrics, Isometric Embeddings in Minkowski Space and the Cross Curvature Flow
75(23)
P. Bryan
M. Ivaki
J. Scheuer
3.1 Introduction
75(2)
3.2 Geometrisation of Three-Manifolds
77(2)
3.3 Embeddability and Hyperbolic Metrics
79(6)
3.4 The Cross Curvature Flow
85(13)
3.4.1 Definition and Basic Properties of the Flow
85(2)
3.4.2 Short Time Existence and Uniqueness
87(3)
3.4.3 Basic Identities and Evolution Equations
90(2)
3.4.4 Towards Hyperbolic Convergence
92(2)
3.4.5 Harnack Inequality and Solitons
94(1)
3.4.6 Monotonicity of Einstein Volume
95(1)
References
96(2)
4 A Mean Curvature Flow for Conformally Compact
98(20)
Manifolds A.R. Gover
V.-M. Wheeler
4.1 Introduction
98(3)
4.2 Conforinal Geometry and Hypersurfaces in Conformally Compact Manifolds
101(7)
4.2.1 Conformal Manifolds
102(1)
4.2.2 The Tractor Connection
102(1)
4.2.3 Conformally Compact Manifolds
103(2)
4.2.4 Hypersurfaces
105(1)
4.2.5 A Hypersurface Flow for Conformally Compact Manifolds
106(1)
4.2.6 Boundary Conditions
107(1)
4.2.7 The Flow Problem
107(1)
4.3 The Flow Problem
108(10)
4.3.1 Treating the Flow as a Nonlinear Partial Differential Equation
108(4)
4.3.2 Generalised Mean Curvature Flow in Hyperbolic Space
112(1)
4.3.3 Long Time Existence and Convergence
113(2)
4.3.4 Generalised Mean Curvature Flow in Riemannian Manifolds
115(1)
References
115(3)
5 A Survey on the Ricci Flow on Singular Spaces
118(23)
K. Kroncke
B. Vertman
5.1 Introduction and Geometric Preliminaries
118(4)
5.1.1 Isolated Conical Singularities
119(2)
5.1.2 Ricci de Turck Flow and the Lichnerowicz Laplacian
121(1)
5.2 Existence of the Singular Ricci de Turck Flow
122(5)
5.2.1 Tangential Stability
123(1)
5.2.2 The Existence Result
124(1)
5.2.3 Characterizing Tangential Stability
125(2)
5.3 Stability of the Singular Ricci de Turck Flow
127(1)
5.4 Perelman's Entropies on Singular Spaces
128(4)
5.4.1 The λ-Functional
128(1)
5.4.2 The Ricci Shrinker Entropy
129(1)
5.4.3 The Ricci Expander Entropy
129(3)
5.5 Curvature Quantities Along Singular Ricci de Turck Flow
132(2)
5.5.1 Bounded Ricci Curvature Along Singular Ricci de Turck Flow
132(1)
5.5.2 Positive Scalar Curvature Along Singular Ricci de Turck Flow
133(1)
5.6 Open Questions and Further Research Directions
134(1)
5.7 Appendix: Sobolev and Holder Spaces
134(7)
References
137(4)
Part Two Structures on Manifolds and Mathematical Physics
141(142)
6 Some Open Problems in Sasaki Geometry
143(26)
C. Boyer
H. Huang
E. Legendre
C. Tønnesen-Friedman
6.1 Introduction
143(2)
6.2 Brief Review of Sasaki Geometry
145(5)
6.2.1 The Sasaki Cone
145(2)
6.2.2 The Transverse Holomorphic Structure
147(1)
6.2.3 The Lie Algebra of Killing Potentials
147(3)
6.3 Extremal Sasaki Geometry
150(11)
6.3.1 Transverse Futaki--Mabuchi
150(3)
6.3.2 The Einstein--Hilbert Functional
153(2)
6.3.3 The Sasaki Energy Functional
155(6)
6.4 The Functionals H, Sε on Lens Space Bundles Over Riemann Surfaces
161(8)
6.4.1 Explicit Examples
163(3)
References
166(3)
7 The Prescribed Ricci Curvature Problem for Homogeneous
169(24)
Metrics T. Buttsworth
A. Pulemotov
7.1 Introduction
169(1)
7.2 The Prescribed Ricci Curvature Problem
169(2)
7.3 Compact Homogeneous Spaces
171(14)
7.3.1 The Variational Interpretation
172(1)
7.3.2 Maximal Isotropy
173(1)
7.3.3 The Structure Constants
173(1)
7.3.4 The Scalar Curvature Functional and its Extension
174(2)
7.3.5 Non-Maximal Isotropy: The First Existence Theorem
176(2)
7.3.6 Non-Maximal Isotropy: The Second Existence Theorem
178(1)
7.3.7 The Case of Two Isotropy Summands
179(2)
7.3.8 Homogeneous Spheres
181(2)
7.3.9 Further Examples
183(2)
7.3.10 Ricci Iterations
185(1)
7.4 Open Questions and Non-Compact Homogeneous Spaces
185(8)
7.4.1 The Non-Compact Case
186(1)
7.4.2 Unimodular Lie Groups of Dimension 3
187(3)
References
190(3)
8 Singular Yamabe and Obata Problems
193(22)
A.R. Gover
A.K. Waldron
8.1 Introduction
193(2)
8.2 Background and a Singular Obata Problem
195(6)
8.3 Tractor Calculus for Hypersurface Embeddings
201(10)
8.3.1 The Sphere
209(2)
8.4 Singular Yamabe and Obata Problems
211(4)
References
212(3)
9 Einstein Metrics, Harmonic Forms and Conformally Kahler Geometry
215(26)
C. LeBrun
9.1 Introduction
215(5)
9.2 An Integral Weitzenbock Formula
220(8)
9.3 Some Almost-Kahler Geometry
228(7)
9.4 The Main Theorems
235(6)
References
238(3)
10 Construction of the Supersymmetric Path Integral: A Survey
241(19)
M. Ludewig
10.1 Introduction
241(3)
10.2 First Construction: The Top Degree Functional
244(5)
10.3 Second Construction: The Chern Character
249(4)
10.4 Bismut--Chern Characters, Entire Chains and the Localization Formula
253(7)
References
257(3)
11 Tight Models of de-Rham Algebras of Highly Connected
260(23)
Manifolds L. Schwachhofer
11.1 Introduction
260(3)
11.2 Rational and Weak Equivalence
263(2)
11.3 Poincare DGCAs and DGCAs of Hodge Type
265(5)
11.4 Small Algebras of Hodge Type DGCAs
270(3)
11.5 Tight DGCAs of Highly Connected DGCAs
273(4)
11.6 The Bianchi-Massey Tensor
277(6)
References
280(3)
Part Three Recent Developments in Non-Negative Sectional Curvature
283
12 Fake Lens Spaces and Non-Negative Sectional Curvature
285(6)
S. Goette
M. Kerin
K. Shankar
12.1 Introduction
285(1)
12.2 "Ze Actions on the Family J
286(5)
References
290(1)
13 Collapsed 3-Dimensional Alexandrov Spaces: A Brief Survey
291(20)
F. Galaz-Garcia
L. Guijarro
J. Nunez-Zimbron
13.1 Introduction
291(2)
13.2 Basic Alexandrov Geometry
293(5)
13.3 Three-Dimensional Alexandrov Spaces
298(4)
13.3.1 Geometric 3-Alexandrov Spaces
301(1)
13.3.2 Geometrization of 3-Alexandrov Spaces
301(1)
13.4 Collapsed Three-Dimensional Alexandrov Spaces
302(9)
13.4.1 General Structure Results
302(5)
13.4.2 Geometrization of Sufficiently Collapsed Three-Dimensional Alexandrov Spaces
307(1)
References
308(3)
14 Pseudo-Angle Systems and the Simplicial Gauss--Bonnet--Chern
311(15)
Theorem S. Klaus
14.1 Introduction
311(2)
14.2 The Simplicial Gauss--Bonnet--Chern Theorem
313(3)
14.3 Systems of Pseudo-Angles
316(3)
14.4 Combinatorial Riemannian Manifolds
319(1)
14.5 Simplicial Sectional Curvature
320(2)
14.6 Simplicial Sectional Curvature and the Hopf Conjecture
322(4)
References
325(1)
15 Aspects and Examples on Quantitative Stratification with Lower Curvature Bounds
326(26)
N. Li
15.1 Introduction
326(2)
15.2 Stratification of Singular Sets
328(2)
15.3 Quantitative Stratification
330(5)
15.3.1 Definitions
330(2)
15.3.2 Results
332(3)
15.4 Key Ingredients and Framework
335(8)
15.4.1 Monotonicity Formula and Bad Scales
335(3)
15.4.2 Splitting Theory
338(2)
15.4.3 Dimension Reduction
340(1)
15.4.4 Good-Scale Annuli Covering
341(2)
15.5 Spaces whose Singular Sets are Cantor Sets
343(9)
References
350(2)
16 Universal Covers of Ricci Limit and RCD Spaces
352(21)
J. Pan
G. Wei
16.1 Introduction
352(2)
16.2 Some Properties of Ricci Limit and RCD Spaces
354(5)
16.3 Universal and δ-Covers
359(5)
16.4 Non-Collapsing Ricci Limit Spaces
364(9)
References
369(4)
17 Local and Global Homogeneity for Manifolds of Positive Curvature
373
J.A. Wolf
17.1 Introduction
373(2)
17.2 The Classification for Positive Curvature
375(2)
17.3 Positive Curvature and Isotropy Splitting
377(1)
17.4 The Three Remaining Positive Curvature Cases
378(2)
17.5 Dropping Normality in Positive Curvature
380
References
383
Owen Dearricott is Honorary Research Fellow at La Trobe University, Australia. A Riemannian geometer best known for his work constructing metrics of positive sectional curvature in dimension seven, he was a co-author of a proceedings volume of the 2010 mini-meeting in Differential Geometry at CIMAT, Guanajuato. Wilderich Tuschmann holds the Differential Geometry Professorial Chair at Karlsruhe Institute of Technology, Germany. He is a geometer with research interests in global differential geometry and geometric topology. He co-authored a scientific biography of the Russian mathematician Sofya Kovalevskaya (1993) and Moduli Spaces of Riemannian Metrics (2015). Yuri Nikolayevsky is Associate Professor at La Trobe University, Victoria. He is a differential geometer best known for his work on Osserman manifolds and homogeneous geometry. He has organised numerous geometry workshops in Australia and was the director of the Australian Mathematical Sciences Institute Summer School at La Trobe University in 2020. Thomas Leistner is Associate Professor at the University of Adelaide. He is a differential geometer who works on Lorentzian and pseudo-Riemann geometry, conformal geometry and holonomy theory. Diarmuid Crowley is Associate Professor at the University of Melbourne. He is a differential topologist with special expertise in the classification of 7-manifolds via smooth invariants.
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