Variational Principles of Topology

One service mathematics has rendered the 'Eot moi, ... , si j'avait JU comment en revenir. human race. h has put common sense back je n'y serais point aUe:' Jules Verne where it belongs, 011 the topmost shelf nen to the dusty canister labeUed 'discarded non- The series is divergent; therefore we may be sense'. able to do something with it. Eric T. Bell O. H es viside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other pans and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Texte du rabat

English translation (from the Russian) of a volume first published in 1982. Multidimensional variational problems are famous for their difficulty, and have generated a vast literature; here the author who writes for the benefit of graduate students and other experts concentrates on methods for minim

Contenu

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.