Variational Principles of Topology

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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,$$\tilde L $$).- 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,$$\tilde L $$ )).- 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 Surfaces of Realizing Type.- 14.3. The Proof of the Reifenberg Conjecture Regarding the Existence of a Universal Upper Estimate of the “Complexity” on the Singular Points of Minimal Surfaces of Realizing Type.- §15 Certain Topological Corollaries. Concrete Series of Examples of Globally Minimal Surfaces of Nontrivial Topological Type.- 15.1. Globally Minimal Surfaces Realizing Nontrivial (Co)Cycles in Symmetric Spaces.- 15.2. Compact Symmetric Spaces and Explicit Form of a Geodesic Diffeomorphism.- 15.3. Explicit Computation of the Deformation Coefficient of a Radial Vector Field on a Symmetric Space.- 15.4. An Explicit Formula for the Symmetric Space Geodesic Nullity.- 15.5. Globally Minimal Surfaces of Least Volume (volkX0 = ?k0 in Symmetric Spaces are Symmetric Spaces of Rank 1.- 15.6. Proof of the Classification Theorem for Surfaces of Least Volume in Certain Classical Symmetric Spaces.- 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.- 17.1. Cohomology Algebras of Compact Lie Groups.- 17.2. Subgroups Totally Nonhomologous to Zero.- 17.3. Pontryagin Cycles in Compact Lie Groups.- 17.4. Necessary Results Concerning 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.- 20.1. The Statement of the Classification Theorem.- 20.2. The Case of Spaces of Type II.- 20.3. The Case of Spaces of Type I (Co) Homologie ally Trivial Cartan Models. Properties of the Squaring Map of a Symmetric Space.- 20.4. The Case of Spaces of Type I. Spaces SU(k)/SO(k).- 20.5. The Case of Spaces of Type I. Spaces SU(2m)/Sp(m).- 20.6. The Case of Spaces of Type I. Spaces S21-1 = SO(2l)/SO(2l - 1). Explicit Computation of Cocycles Realizable by Totally Geodesic Submanifolds of Type I.- 20.7. The Case of Spaces of Type I. Space E6/F4.- §21 Classification Theorem Describing Cocycles in the Compact Lie Group Cohomology Realizable by Totally Geodesic Spheres.- 21.1. Classification Theorem Formulation.- 21.2. Totally Geodesic Spheres Realizing Bott Periodicity.- 21.3. Realization of Homotopy Group Elements of the Compact Lie Groups by Totally Geodesic Spheres.- 21.4. Necessary Results Concerning the Spinor and Semispinor Representations of an Orthogonal Group.- 21.5. Spinor Representation of the Orthogonal Group SO(8) and the Cayley Number Automorphism Group.- 21.6. Description of Totally Geodesic Spheres Realizing Nontrivial (Co)Cycles in Simple Lie Group Cohomology. The Case of the Group SU(n).- 21.7. The Case of the Groups SO(n) and Sp(2n).- §22 Classification Theorem Describing Elements of Homotopy Groups of Symmetric Spaces of Type I, Realizable by Totally Geodesic Spheres.- 22.1. Classification Theorem Statement.- 22.2. Proof of the Classification Theorem. Relation between the Number of Linearly Independent Fields on Spheres and that of the Elements of Homotopy Groups Realizable by Totally Geodesic Spheres.- 5. Variational Methods for Certain Topological Problems.- §23 Bott Periodicity from the Dirichlet Multidimensional Functional Standpoint.- 23.1. Explicit Description of the Bott Periodicity Isomorphism for the Unitary Group.- 23.3. Unitary Periodicity and One-Dimensional Functional.- 23.4. The Periodicity Theorem for a Unitary Group is Based on the Dirichlet Functional Two-Dimensional Extremals.- 23.5. The Periodicity Theorem for an Orthogonal Group is Based on the 8-Dimensional Dirichlet Functional Extremals.- §24 Three Geometric Problems of Variational Calculus.- 24.1. Minimal Cones and Singular Points of Minimal Surfaces.- 24.2. The Equivariant Plateau Problem.- 24.3. Representation of Equivariant Singularities as Singular Points of Closed Minimal Surfaces Embedded into Symmetric Spaces.- 24.4. On the Existence of Nonlinear Functions Whose Graphs in Euclidean Space Are Minimal Surfaces.- 24.5. Harmonic Mappings of Spheres in Nontrivial Homotopy Classes.- 24.6. A Rough Outline of Certain Recent Results on the Link of Harmonic Mapping Properties to the Topology of Manifolds.- 24.7. Properties of the Density of Smooth Mappings of Manifolds.- 24.8. The Behaviour of the Dirichlet Functional on the 2-Connected Manifold Diffeomorphism Group. Proof of Theorem 24.6.9.- 24.9. Necessary Topological Condition for the Existence of Nontrivial Globally Minimal Harmonic Mappings.- 24.10. The Minimization of Dirichlet-Type Functionals.- 24.11. Regularity of Harmonic Mappings.- 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).- §27 Closedness, Invariance, and Stability of Variational Classes.- 27.1. S-Surgery of Surfaces in a Riemannian Manifold.- 27.2. The Closedness of Variational Classes Relative to the Passage to the Limit.- 27.3. The Invariance of Variational Classes Relative to S-Surgeries of Surfaces.- 27.4. The Stability of Variational Classes.- §28 The General Isoperimetric Inequality.- 28.1. Choice of a Special Coordinate System.- 28.2. Simplicial Points of Surfaces.- 28.3. Isoperimetric Inequality.- §29 The Minimizing Process in Variational Classes and h(A,L,$$\tilde L $$ ).- 29.1. The Minimizing Sequence of Surfaces. Density Functions Related to Surfaces.- 29.2. A Rough Outline of the Minimizing Process.- 29.3. The Constructive Method for the Minimizing Process and the Proof for Its Convergence. First Step.- 29.4. Second and Subsequent Steps in the Minimizing Process.- 29.5. The Theorem on the Coincidence of the Least Stratified Volume with Least ?-Vector in a Variational Class.- §30 Properties of Density Functions. The Minimality of Each Stratum of the Surface Obtained in the Minimization Process.- 30.1. The Value of the Density Function is Always not Less Than Unity on Each Stratum, and Unity only at Regular Points.- 3.2. Each Stratum Is a Smooth Minimal Submanifold, Except Possibly a Set of Singular Points of Measure Zero.- §31 Proof of Global Minimality for Constructed Stratified Surfaces.- 31.1. Proof of the Basic Existence Theorem for a Globally Minimal Surface.- 31.2. The Proof of the Theorem on the Coincidence of the Least Stratified Volume with the Least ?-Vector in a Variational Class.- §32 The Fundamental (Co)Cycles of Globally Minimal Surfaces. Exact Realization and Exact Spanning.- 32.1. Fundamental (Co)Cycle Theorem.- 32.2. Exact Minimal Realization and Exact Minimal Spanning.- 32.3. Minimal Surfaces with Boundaries Homeomorphic to the Sphere.- 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.