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Variational Principles of Topology: Multidimensional Minimal Surface Theory
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(Buch) |
Dieser Artikel gilt, aufgrund seiner Grösse, beim Versand als 3 Artikel!
| Inhalt: |
| 1. Simplest Classical Variational Problems.- ?Equations of Extremals for Functionals.- ?Geometry of Extremals.- 2. Multidimensional Variational Problems and Extraordinary (Co)Homology Theory.- ?The Multidimensional Plateau Problem and Its Solution in the Class of Mapping on Spectra of Manifolds with Fixed Boundary.- ?Extraordinary (Co)Homology Theories Determined for "Surfaces with Singularities".- ?The Coboundary and Boundary of a Pair of Spaces (X, A).- ?Determination of Classes of Admissible Variations of Surfaces in Terms of (Co)Boundary of the Pair(X, A).- ?Solution of the Plateau Problem (Finding Globally Minimal Surfaces (Absolute Minimum) in the Variational Classes h(A,L,L?) and h(A,$$\tilde L $$
)).- ?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.- ?Exhaustion Functions and Minimal Surfaces.- ? Definition and Simplest Properties of the Deformation Coefficient of a Vector Field.- ? Formulation of the Basic Theorem for the Lower Estimate of the Minimal Surface Volume Function.- ? Proof of the Basic Volume Estimation Theorem.- ? Certain Geometric Consequences.- ? Nullity of Riemannian, Compact, and Closed Manifolds. Geodesic Nullity and Least Volumes of Globally Minimal Surfaces of Realizing Type.- ? 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.- ? Problem Formulation. Totally Geodesic Submanifolds in Lie Groups.- ? Necessary Results Concerning the Topological Structure of Compact Lie Groups and Symmetric Spaces.- ? Lie Groups Containing a Totally Geodesic Submanifold Necessarily Contain Its Isometry Group.- ? 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.- ? Classification Theorem Describing Totally Geodesic Submanifolds Realizing Nontrivial (Co)Cycles in Compact Lie Group (Co) Homology.- ? Classification Theorem Describing Cocycles in the Compact Lie Group Cohomology Realizable by Totally Geodesic Spheres.- ? 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.- ? Bott Periodicity from the Dirichlet Multidimensional Functional Standpoint.- ? 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, $$\tilde L $$
)).- ? The Cohomology Case. Computation of the Coboundary of the Pair (X,A) = ?r(Xr,Ar) in Terms of Those of (Xr,Ar).- ? The Homology Case. Computation of the Boundary of the Pair (X,A) = ?r(Xr,Ar) in Terms of the Boundaries of (Xr,Ar).- ? The General Isoperimetric Inequality.- ? The Minimizing Process in Variational Classes and h(A,L,$$\tilde L $$
).- ? Properties of Density Functions. The Minimality of Each Stratum of the Surface Obtained in the Minimization Process.- ? Proof of Global Minimality for Constructed Stratified Surfaces.- ? The Fundamental (Co)Cycles of Globally Minimal Surfaces. Exact Realization and Exact Spanning.- Appendix I. Minimality Test for Lagrangian Submanifolds in K?ler Manifolds. Submanifolds in K?ler Manifolds. Maslov Index in Minimal Surface Theory.- ?Definitions.- ?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-Dimensio |
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