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E-grāmata: Minimal Submanifolds In Pseudo-riemannian Geometry

(Univ Of Sao Paulo, Brazil)
  • Formāts: 184 pages
  • Izdošanas datums: 02-Nov-2010
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789814466141
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Minimal Submanifolds In Pseudo-riemannian GeometryPalielināt
  • Formāts: 184 pages
  • Izdošanas datums: 02-Nov-2010
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789814466141
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Since the foundational work of Lagrange on the differential equation to be satisfied by a minimal surface of the Euclidean space, the theory of minimal submanifolds have undergone considerable developments, involving techniques from related areas, such as the analysis of partial differential equations and complex analysis. On the other hand, the relativity theory has led to the study of pseudo-Riemannian manifolds, which turns out to be the most general framework for the study of minimal submanifolds. However, most of the recent books on the subject still present the theory only in the Riemannian case.For the first time, this book provides a self-contained and accessible introduction to the subject in the general setting of pseudo-Riemannian geometry, only assuming from the reader some basic knowledge about manifold theory. Several classical results, such as the Weierstrass representation formula for minimal surfaces, and the minimizing properties of complex submanifolds, are presented in full generality without sacrificing the clarity of exposition. Finally, a number of very recent results on the subject, including the classification of equivariant minimal hypersurfaces in pseudo-Riemannian space forms and the characterization of minimal Lagrangian surfaces in some pseudo-Kähler manifolds are given.
Foreword vii
Preface ix
1 Submanifolds in pseudo-Riemannian geometry
1(28)
1.1 Pseudo-Riemannian manifolds
1(8)
1.1.1 Pseudo-Riemannian metrics
1(2)
1.1.2 Structures induced by the metric
3(5)
1.1.3 Calculus on a pseudo-Riemannian manifold
8(1)
1.2 Submanifolds
9(9)
1.2.1 The tangent and the normal spaces
9(2)
1.2.2 Intrinsic and extrinsic structures of a submanifold
11(3)
1.2.3 One-dimensional submanifolds: Curves
14(3)
1.2.4 Submanifolds of co-dimension one: Hypersurfaces
17(1)
1.3 The variation formulae for the volume
18(9)
1.3.1 Variation of a submanifold
18(1)
1.3.2 The first variation formula
19(4)
1.3.3 The second variation formula
23(4)
1.4 Exercises
27(2)
2 Minimal surfaces in pseudo-Euclidean space
29(28)
2.1 Intrinsic geometry of surfaces
29(3)
2.2 Graphs in Minkowski space
32(8)
2.3 The classification of ruled, minimal surfaces
40(7)
2.4 Weierstrass representation for minimal surfaces
47(7)
2.4.1 The definite case
48(4)
2.4.2 The indefinite case
52(2)
2.4.3 A remark on the regularity of minimal surfaces
54(1)
2.5 Exercises
54(3)
3 Equivariant minimal hypersurfaces in space forms
57(32)
3.1 The pseudo-Riemannian space forms
57(4)
3.2 Equivariant minimal hypersurfaces in pseudo-Euclidean space
61(8)
3.2.1 Equivariant hypersurfaces in pseudo-Euclidean space
61(2)
3.2.2 The minimal equation
63(2)
3.2.3 The definite case (ε,ε') = (1,1)
65(1)
3.2.4 The indefinite positive case (ε,ε') = (-1,1)
66(1)
3.2.5 The indefinite negative case (ε,ε') = (-1,-1)
67(1)
3.2.6 Conclusion
68(1)
3.3 Equivariant minimal hypersurfaces in pseudo-space forms
69(17)
3.3.1 Totally umbilic hypersurfaces in pseudo-space forms
69(4)
3.3.2 Equivariant hypersurfaces in pseudo-space forms
73(2)
3.3.3 Totally geodesic and isoparametric solutions
75(1)
3.3.4 The spherical case (ε,ε',ε") = (1,1,1)
76(2)
3.3.5 The "elliptic hyperbolic" case (ε,ε',ε") = (1,-1,-1)
78(2)
3.3.6 The "hyperbolic hyperbolic" case (ε,ε',ε") = (-1,-1,1)
80(1)
3.3.7 The "elliptic" de Sitter case (ε,ε',ε") = (-1,1,1)
81(1)
3.3.8 The "hyperbolic" de Sitter case (ε,ε',ε") = (1,-1,1)
82(2)
3.3.9 Conclusion
84(2)
3.4 Exercises
86(3)
4 Pseudo-Kahler manifolds
89(22)
4.1 The complex pseudo-Euclidean space
89(2)
4.2 The general definition
91(4)
4.3 Complex space forms
95(5)
4.3.1 The case of dimension n = 1
99(1)
4.4 The tangent bundle of a psendo-Kahler manifold
100(9)
4.4.1 The canonical symplectic structure of the cotangent bundle TM
100(2)
4.4.2 An almost complex structure on the tangent bundle TM of a manifold equipped with an affine connection
102(2)
4.4.3 Identifying TM and TM and the Sasaki metric
104(2)
4.4.4 A complex structure on the tangent bundle of a pseudo-Kahler manifold
106(2)
4.4.5 Examples
108(1)
4.5 Exercises
109(2)
5 Complex and Lagrangian submanifolds in pseudo-Kahler manifolds
111(40)
5.1 Complex submanifolds
111(2)
5.2 Lagrangian submanifolds
113(1)
5.3 Minimal Lagrangian surfaces in C2 with neutral metric
114(2)
5.4 Minimal Lagrangian submanifolds in Cn
116(11)
5.4.1 Lagrangian graphs
118(2)
5.4.2 Equivariant Lagrangian submanifolds
120(3)
5.4.3 Lagrangian submanifolds from evolving quadrics
123(4)
5.5 Minimal Lagrangian submanifols in complex space forms
127(16)
5.5.1 Lagrangian and Legendrian submanifolds
128(5)
5.5.2 Equivariant Legendrian submanifolds in odd-dimensional space forms
133(4)
5.5.3 Minimal equivariant Lagrangian submanifolds in complex space forms
137(6)
5.6 Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface
143(5)
5.6.1 Rank one Lagrangian surfaces
144(2)
5.6.2 Rank two Lagrangian surfaces
146(2)
5.7 Exercises
148(3)
6 Minimizing properties of minimal submanifolds
151(10)
6.1 Minimizing submanifolds and calibrations
151(7)
6.1.1 Hypersurfaces in pseudo-Euclidean space
151(4)
6.1.2 Complex submanifolds in pseudo-Kahler manifolds
155(1)
6.1.3 Minimal Lagrangian submanifolds in complex pseudo-Euclidean space
156(2)
6.2 Non-minimizing submanifolds
158(3)
Bibliography 161(4)
Index 165
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