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Minimal submanifolds in pseudo-Riemannian geometry

Author: Henri Anciaux
Publisher: Singapore ; Hackensack, NJ : World Scientific, 2011.
Edition/Format:   Print book : EnglishView all editions and formats
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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  Read more...

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Document Type: Book
All Authors / Contributors: Henri Anciaux
ISBN: 9789814291248 9814291242
OCLC Number: 700137424
Description: xv, 167 pages : illustrations ; 24 cm
Contents: Machine generated contents note: 1. Submanifolds in pseudo-Riemannian geometry --
1.1. Pseudo-Riemannian manifolds --
1.1.1. Pseudo-Riemannian metrics --
1.1.2. Structures induced by the metric --
1.1.3. Calculus on a pseudo-Riemannian manifold --
1.2. Submanifolds --
1.2.1. The tangent and the normal spaces --
1.2.2. Intrinsic and extrinsic structures of a submanifold --
1.2.3. One-dimensional submanifolds: Curves --
1.2.4. Submanifolds of co-dimension one: Hypersurfaces --
1.3. The variation formulae for the volume --
1.3.1. Variation of a submanifold --
1.3.2. The first variation formula --
1.3.3. The second variation formula --
1.4. Exercises --
2. Minimal surfaces in pseudo-Euclidean space --
2.1. Intrinsic geometry of surfaces --
2.2. Graphs in Minkowski space --
2.3. The classification of ruled, minimal surfaces --
2.4. Weierstrass representation for minimal surfaces --
2.4.1. The definite case --
2.4.2. The indefinite case --
2.4.3.A remark on the regularity of minimal surfaces --
2.5. Exercises --
3. Equivariant minimal hypersurfaces in space forms --
3.1. The pseudo-Riemannian space forms --
3.2. Equivariant minimal hypersurfaces in pseudo-Euclidean space --
3.2.1. Equivariant hypersurfaces in pseudo-Euclidean space --
3.2.2. The minimal equation --
3.2.3. The definite case (& epsilon;, & epsilon;') = (1,1) --
3.2.4. The indefinite positive case (& epsilon;, & epsilon;') = ( -1,1) --
3.2.5. The indefinite negative case (& epsilon;, & epsilon;') = ( -1,-1) --
3.2.6. Conclusion --
3.3. Equivariant minimal hypersurfaces in pseudo-space forms --
3.3.1. Totally umbilic hypersurfaces in pseudo-space forms --
3.3.2. Equivariant hypersurfaces in pseudo-space forms --
3.3.3. Totally geodesic and isoparametric solutions --
3.3.4. The spherical case (& epsilon;, & epsilon;', & epsilon;") = (1,1,1) --
3.3.5. The "elliptic hyperbolic" case (& epsilon;, & epsilon;', & epsilon;") = (1,-1,-1) --
3.3.6. The "hyperbolic hyperbolic" case (& epsilon;, & epsilon;', & epsilon;") = ( -1,-1,1) --
3.3.7. The "elliptic" de Sitter case (& epsilon;, & epsilon;', & epsilon;") = ( -1,1,1) --
3.3.8. The "hyperbolic" de Sitter case (& epsilon;, & epsilon;', & epsilon;") = (1,-1,1) --
3.3.9. Conclusion --
3.4. Exercises --
4. Pseudo-Kahler manifolds --
4.1. The complex pseudo-Euclidean space --
4.2. The general definition --
4.3.Complex space forms --
4.3.1. The case of dimension n = 1 --
4.4. The tangent bundle of a psendo-Kahler manifold --
4.4.1. The canonical symplectic structure of the cotangent bundle TM --
4.4.2. An almost complex structure on the tangent bundle TM of a manifold equipped with an affine connection --
4.4.3. Identifying TM and TM and the Sasaki metric --
4.4.4.A complex structure on the tangent bundle of a pseudo-Kahler manifold --
4.4.5. Examples --
4.5. Exercises --
5.Complex and Lagrangian submanifolds in pseudo-Kahler manifolds --
5.1.Complex submanifolds --
5.2. Lagrangian submanifolds --
5.3. Minimal Lagrangian surfaces in C2 with neutral metric --
5.4. Minimal Lagrangian submanifolds in Cn --
5.4.1. Lagrangian graphs --
5.4.2. Equivariant Lagrangian submanifolds --
5.4.3. Lagrangian submanifolds from evolving quadrics --
5.5. Minimal Lagrangian submanifols in complex space forms --
5.5.1. Lagrangian and Legendrian submanifolds --
5.5.2. Equivariant Legendrian submanifolds in odd-dimensional space forms --
5.5.3. Minimal equivariant Lagrangian submanifolds in complex space forms --
5.6. Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface --
5.6.1. Rank one Lagrangian surfaces --
5.6.2. Rank two Lagrangian surfaces --
5.7. Exercises --
6. Minimizing properties of minimal submanifolds --
6.1. Minimizing submanifolds and calibrations --
6.1.1. Hypersurfaces in pseudo-Euclidean space --
6.1.2.Complex submanifolds in pseudo-Kahler manifolds --
6.1.3. Minimal Lagrangian submanifolds in complex pseudo-Euclidean space --
6.2. Non-minimizing submanifolds.
Responsibility: Henri Anciaux.

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The book is written in an accessible and quite self-contained way. It is recommendable for a broad audience of students and mathematicians interested in minimal submanifolds in pseudo-Riemannian Read more...

 
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