Variational Principles of Topology: Multidimensional Minimal Surface Theory(42 Mathematics and its Applications)

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Table of Contents:
1. Simplest Classical Variational Problems.- §1 Equations of Extremals for Functionals.- §2 Geometry of Extremals.- 2. Multidimensional Variational Problems and Extraordinary (Co)Homology Theory.- §3 The Multidimensional Plateau Problem and Its Solution in the Class of Mapping on Spectra of Manifolds with Fixed Boundary.- §4 Extraordinary (Co)Homology Theories Determined for “Surfaces with Singularities”.- §5 The Coboundary and Boundary of a Pair of Spaces (X, A).- §6 Determination of Classes of Admissible Variations of Surfaces in Terms of (Co)Boundary of the Pair(X, A).- §7 Solution of the Plateau Problem (Finding Globally Minimal Surfaces (Absolute Minimum) in the Variational Classes h(A,L,L?) and h(A,$$\tilde L $$ )).- §8 Solution of the Problem of Finding Globally Minimal Surfaces in Each Homotopy Class of Multivarifolds.- 3. Explicit Calculation of Least Volumes (Absolute Minimum) of Topologically Nontrivial Minimal Surfaces.- §9 Exhaustion Functions and Minimal Surfaces.- §10 Definition and Simplest Properties of the Deformation Coefficient of a Vector Field.- §11 Formulation of the Basic Theorem for the Lower Estimate of the Minimal Surface Volume Function.- §12 Proof of the Basic Volume Estimation Theorem.- §13 Certain Geometric Consequences.- §14 Nullity of Riemannian, Compact, and Closed Manifolds. Geodesic Nullity and Least Volumes of Globally Minimal Surfaces of Realizing Type.- §15 Certain Topological Corollaries. Concrete Series of Examples of Globally Minimal Surfaces of Nontrivial Topological Type.- 4. Locally Minimal Closed Surfaces Realizing Nontrivial (Co)Cycies and Elements of Symmetric Space Homotopy Groups.- §16 Problem Formulation. Totally Geodesic Submanifolds in Lie Groups.- §17 Necessary Results Concerning the Topological Structure of Compact Lie Groups and Symmetric Spaces.- §18 Lie Groups Containing a Totally Geodesic Submanifold Necessarily Contain Its Isometry Group.- §19 Reduction of the Problem of the Description of (Co)Cycles Realizable by Totally Geodesic Submanifolds to the Problem of the Description of (Co)Homological Properties of Cartan Models.- §20 Classification Theorem Describing Totally Geodesic Submanifolds Realizing Nontrivial (Co)Cycles in Compact Lie Group (Co) Homology.- §21 Classification Theorem Describing Cocycles in the Compact Lie Group Cohomology Realizable by Totally Geodesic Spheres.- §22 Classification Theorem Describing Elements of Homotopy Groups of Symmetric Spaces of Type I, Realizable by Totally Geodesic Spheres.- 5. Variational Methods for Certain Topological Problems.- §23 Bott Periodicity from the Dirichlet Multidimensional Functional Standpoint.- §24 Three Geometric Problems of Variational Calculus.- 6. Solution of the Plateau Problem in Classes of Mappings of Spectra of Manifolds with Fixed Boundary. Construction of Globally Minimal Surfaces in Variational Classes h(A,L, L?) and h(A, $$\tilde L $$ )).- §25 The Cohomology Case. Computation of the Coboundary of the Pair (X,A) = ?r(Xr,Ar) in Terms of Those of (Xr,Ar).- §26 The Homology Case. Computation of the Boundary of the Pair (X,A) = ?r(Xr,Ar) in Terms of the Boundaries of (Xr,Ar).- §28 The General Isoperimetric Inequality.- §29 The Minimizing Process in Variational Classes and h(A,L,$$\tilde L $$ ).- §30 Properties of Density Functions. The Minimality of Each Stratum of the Surface Obtained in the Minimization Process.- §31 Proof of Global Minimality for Constructed Stratified Surfaces.- §32 The Fundamental (Co)Cycles of Globally Minimal Surfaces. Exact Realization and Exact Spanning.- Appendix I. Minimality Test for Lagrangian Submanifolds in Kähler Manifolds. Submanifolds in Kähler Manifolds. Maslov Index in Minimal Surface Theory.- §1 Definitions.- §3 Certain Corollaries. New Examples of Minimal Surfaces. The Maslov Index for Minimal Lagrangian Submanifolds.- Appendix II. Calibrations, Minimal Surface Indices, Minimal Cones of Large Codimensional and the One-Dimensional Plateau Problem.