Riemannian Topology and Geometric Structures on Manifolds (eBook)

Preface 8
Charles P. Boyer — An Autobiographical Sketch 10
Charles P. Boyer’s List of Publications 10
Accepted for Publication 15
Contents 16
L2-Cohomology of Spaces with Nonisolated Conical Singularities and Nonmultiplicativity of the Signature 17
1 Introduction 17
2 Reviewof L2-Cohomology 23
3 L2-Cohomology of Generalized Thom Spaces 25
4 Spectral Sequences 28
5 The Generalized Thom Isomorphism 29
6 L2-Signatures of Generalized Thom Spaces 34
7 L2-Cohomology of Hyperkähler Manifolds 38
References 39
Hirzebruch Surfaces and Weighted Projective Planes 41
1 The Bochner-Flat Metric of Weighted Projective Spaces 42
2 Calabi Extremal Kähler Metrics on Hirzebruch Surfaces 46
3 Hirzebruch Surfaces as Toric Kähler Manifolds 52
4 TheWeighted 2-Plane P2k as a Blow- Down of Fk 59
References 63
Quaternionic Kähler Moduli Spaces 65
1 Introduction 65
2 Special Kähler Manifolds 66
3 The Hyperkähler Manifold 67
4 The Heisenberg Group 70
5 Quaternionic Structures 71
6 Taking the Quotient 73
7 Properties of the Metric 76
References 77
Homological Mirror Symmetry and Algebraic Cycles 79
1 Introduction 79
2 Definitions 80
3 The Construction 83
4 Birational Transformations, Earthquakes, and HMS 86
5 Homological Mirror Symmetry and Tropical Hodge Conjecture 102
References 107
Positive Sasakian Structures on 5-Manifolds 109
1 Reduction to Algebraic Geometry 114
2 The Exceptional Cases 118
3 The Main Series 124
4 Klt1 Conditions 130
References 132
Four-Manifolds, Curvature Bounds, and Convex Geometry 135
1 Introduction 135
2 Rudiments of 4-Dimensional Geometry 137
3 The Seiberg–Witten Equations 142
4 Monopoles and Convex Hulls 152
5 Theory and Practice 159
References 167
The 1-Nullity of Sasakian Manifolds 169
1 Introduction 169
2 Preliminaries 169
3 The k-Nullity Distribution 170
4 The 1-Nullity of Sasakian Manifolds 172
References 175
New Results in Sasaki–Einstein Geometry 177
1 Introduction 177
2 Sasakian Geometry 179
3 Explicit Constructions of Sasaki–Einstein Manifolds 181
4 Toric Sasakian Geometry 183
5 A Variational Problem for the Reeb Vector Field 187
6 Obstructions to the Existence of Sasaki–Einstein Metrics 193
References 199
Some Examples of Toric Sasaki–Einstein Manifolds 201
1 Introduction 201
2 Sasakian Manifolds 203
3 Toric Geometry 205
4 Einstein Equation 223
5 Three-Sasakian Manifolds 226
6 Sasakian Submanifolds 236
7 Positive Ricci Curvature Examples 243
8 Higher Dimensional Examples 245
References 247
On the Geometry of Cohomogeneity One Manifolds with Positive Curvature 249
1 Preliminaries 250
2 Known Examples of Cohomogeneity One Manifolds with Positive Curvature 252
3 Classification of Cohomogeneity One Manifolds with Positive Curvature 264
4 Candidates and Hitchin Metrics 269
References 276
The Sasaki Cone and Extremal Sasakian Metrics 279
1 Introduction 279
2 Preliminaries on Sasakian Geometry 281
3 The Energy Functional in the Sasaki Cone 285
4 The Euler–Lagrange Equation for Strongly Extremal Reeb Vector Fields 286
5 Sasakian Geometry of Links of Isolated Hypersurface Singularities 289
6 Extremal Sasakian Metrics in Dimension Five 294
7 Appendix 300
References 304