Contents -- Foreword -- Preface -- Chapter 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 -- 1.1.2.1 Volume -- 1.1.2.2 The Levi-Civita connection -- 1.1.2.3 Curvature of a connection -- 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 -- 1.2.3.1 Arc length -- 1.2.3.2 Curvature of a curve -- 1.2.3.3 Curves in surfaces and the Frénet equations -- 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 -- Chapter 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 -- 2.4.1.1 The case of dimension 3 -- 2.4.2 The indefinite case -- 2.4.3 A remark on the regularity of minimal surfaces -- 2.5 Exercises -- Chapter 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 (ε, ε') = (1, 1) -- 3.2.4 The indefinite positive case (ε, ε') = (−1, 1) -- 3.2.5 The indefinite negative case (ε, ε') = (−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 (ε, ε', ε") = (1, 1, 1) -- 3.3.5 The "elliptic hyperbolic" case (ε, ε', ε") = (1,−1,−1) -- 3.3.6 The "hyperbolic hyperbolic" case (ε, ε', ε") = (−1,−1, 1) -- 3.3.7 The "elliptic" de Sitter case (ε, ε', ε") = (−1, 1, 1) -- 3.3.7.1 The positive case -- 3.3.8 The "hyperbolic" de Sitter case (ε, ε', ε") = (1,−1, 1) -- 3.3.8.1 The region {

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 -- Chapter 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-Kähler manifolds -- 6.1.3 Minimal Lagrangian submanifolds in complex pseudo-Euclidean space -- 6.2 Non-minimizing submanifolds -- Bibliography -- Index.