Variational Principles of Topology: Multidimensional Minimal Surface Theory (Paperback, 1990)

1. Simplest Classical Variational Problems.- 1 Equations of Extremals for Functionals.- 2 Geometry of Extremals.- 2.1. The Zero-Dimensional and One-Dimensional Cases.- 2.2. Some Examples of the Simplest Multidimensional Functional. The Volume Functional.- 2.3. The Classical Plateau Problem in Dimension 2.- 2.4. The Second Fundamental Form on the Riemannian Submanifold.- 2.5. Local Minimality.- 2.6. First Examples of Globally Minimal Surfaces.- 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.- 3.1. The Classical Formulations (Finding the Absolute Minimum).- 3.2. The Classical Formulations (Finding a Relative Minimum).- 3.3. Difficulties Arising in the Minimization of the Volume Functional volk for k 2. Appearance on Nonremovable Strata of Small Dimensions.- 3.4. Formulations of the Plateau Problem in the Language of the Usual Spectral Homology.- 3.5. The Classical Multidimensional Plateau Problem (the Absolute Minimum) and the Language of Bordism Theory.- 3.6. Spectral Bordism Theory as an Extraordinary Homology Theory.- 3.7. The Formulation of the Solution to the Plateau Problem (Existence of the Absolute Minimum in Spectral Bordism Classes).- 4 Extraordinary (Co)Homology Theories Determined for "Surfaces with Singularities".- 4.1. The Characteristic Properties of (Co)Homology Theories.- 4.2. Extraordinary (Co)Homology Theories for Finite Cell Complexes.- 4.3. The Construction of Extraordinary (Co)Homology Theories for "Surfaces with Singularities" (on Compact Sets).- 4.4. Verifying the Characteristic Properties of the Constructed Theories.- 4.5. Additional Properties of Extraordinary Spectral Theories.- 4.6. Reduced (Co)Homology Groups on "Surfaces with Singularities".- 5 The Coboundary and Boundary of a Pair of Spaces (X, A).- 5.1. The Coboundary of a Pair (X,A).- 5.2. The Boundary of a Pair (X,A).- 6 Determination of Classes of Admissible Variations of Surfaces in Terms of (Co)Boundary of the Pair(X, A).- 6.1. Variational Classes h(A,L,L?) and h(A,$$ ilde L $$).- 6.2. The Stability of Variational Classes.- 7 Solution of the Plateau Problem (Finding Globally Minimal Surfaces (Absolute Minimum) in the Variational Classes h(A,L,L?) and h(A,$$ ilde L $$ )).- 7.1. The Formulation of the Problem.- 7.2. The Basic Existence Theorem for Globally Minimal Surfaces. Solution of the Plateau Problem.- 7.3. A Rough Outline of the Existence Theorem.- 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.- 9.1. Certain Classical Problems.- 9.2. Bordisms and Exhaustion Functions.- 9.3. GM-Surfaces.- 9.4. Formulation of the Problem of a Lower Estimate of the Minimal Surface Volume Function.- 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.- 11.1. Functions of the Interaction of a Globally Minimal Surface with a Wavefront.- 11.2. Formulation of the Basic Volume Estimation Theorem.- 12 Proof of the Basic Volume Estimation Theorem.- 13 Certain Geometric Consequences.- 13.1. On the Least Volume of Globally Minimal Surfaces Passing through the Centre of a Ball in Euclidean Space.- 13.2. On the Least Volume of Globally Minimal Surfaces Passing through a Fixed Point in a Manifold.- 13.3. On the Least Volume of Globally Minimal Surfaces Formed by the Integral Curves of a Field ?.- 14 Nullity of Riemannian, Compact, and Closed Manifolds. Geodesic Nullity and Least Volumes of Globally Minimal Surfaces of Realizing Type.- 14.1. The Definition of the Nullity of a Manifold.- 14.2. The Theorem on the Relation of Nullity with the Least Volumes of Sur