The author demonstrates the deep rooted connections between the theory of minimal surfaces and modern branches of mathematics, particularly the theory of differential equations, Lie groups and multidimensional variational calculus.
Explores the history and current state of the theory of minimal surfaces containing original research data.
Its clear presentation and numerous illustrations make this topic accessible to both students and research workers in the fields of mathematics and physics.
The Plateau Problem
Part I Historical Survey
Part II Present State of the Theory
List of Contents of Part II
IV MODERN STATE OF MINIMAL SURFACE THEORY
§1. Minimal Surfaces and Homology.
1. Two-dimensional minimal surfaces in Euclidean space and in Riemannian manifolds.
2. The multidimensional Riemannian volume functional and second fundamental form on a submanifold.
3. Multidimensional locally minimal surfaces.
4. The global minimality of complex submanifolds.
5. The complex Plateau problem.
6. On various approaches to the concepts of surface and boundary of a surface.
7. The homology boundary of a surface and the role of the coefficient group.
8. Surprising examples of physical stable minimal surfaces, that nevertheless retract onto their boundaries.
9. When does a soap film spanning a frame not contain closed soap-bubbles?
§2. Integral Currents.
1. De Rham currents. Basic notions.
2. Rectifiable currents and flat chains.
3. Normal and integral currents.
4. Various formulations of the minimal current existence theorem.
5. Varifolds and minimal surfaces
6. The interior regularity for minimal surfaces and the structure of their singular points.
7. Regularity almost everywhere for the supports of elliptic-integrand-minimizing k-currents and k-varifolds.
8. The interior regularity for volume-minimizing hypersurfaces and the existence of minimal cones of codimension one.
9. Dimension estimates for the set of singular points of a minimal surface.
10. Other problems of minimal surface regularity.
§3. Minimal Currents in Riemannian Manifolds.
1. Minimal cones associated with singular points of minimal surfaces.
2. Multidimensional minimal cones.
3. Minimal surfaces invariant with respect to the action of Lie groups.
4. The Fermat principle, minimal cones, and light rays.
5. S.N. Bernstein's problem.
§4. Minimization of Volumes of Manifolds with Fixed Boundary and of Closed Manifolds. Existence of a Minimum in Each Spectral Bordism Class.
1. Bordant manifolds and the multidimensional Plateau problem.
2. The properties of bordant manifold classes.
3. The statement of the existence theorem for globally minimal surfaces in the spectral bordant manifold class.
§5. Generalized Homology and Cohomology Theories and Their Relation to the Multidimensional Plateau Problem.
1. The definition of generalized homology and cohomology.
2. The coboundary and boundary of a pair of spaces (X,A).
3. Surface variational classes.
4. The general existence theorem for globally minimal surfaces in an arbitrary class determined by a generalized spectral homology or cohomology theory.
5. A short sketch of the proof of Theorem 1
§6. Existence of a Minimum in Each Homotopy Class of Multivarifolds.
1. The functional multivarifold language.
2. Multivarifolds and variational problems in the classes of surfaces of fixed topological type.
3. Minimization problems for generalized integrands in the parametrization and parametrized multivarifold classes.
4. Criteria for the global minimality of surfaces and currents.
§7. Cases where a Solution of the Dirichlet Problem for the Equation of Minimal Surfaces of High Codimensions does not Exist.
§8. Example of a Smooth, Closed, Unknotted Curve in R3 , Bounding Only Minimal surfaces of Large Genus.
§9. Certain New Methods of Effective Construction of Globally Minimal Surfaces in Riemannian Manifolds.
1. The universal lower estimate of the volumes of topologically non-trivial minimal surfaces.
2. The coefficient of deformation of a vector field.
3. Surfaces of non-trivial topological type and of least volume.
4. On the minimal volume of surfaces passing through the centre of a symmetric convex domain in Euclidean space.
§10. Totally Geodesic Surfaces Realizing Non-Trivial Cycles, Cocycles, and Elements of Homotopy Groups in Symmetric Spaces.
1. Totally geodesic submanifolds in Lie groups.
2. Necessary information about symmetric spaces.
3. When does a totally geodesic submanifold realize a nontrivial cycle?
4. The classification theorem for totally geodesic submanifolds realizing non-trivial cycles in symmetric spaces.
5. The classification of cocycles realizable by totally geodesic spheres in compact Lie groups.
6. The classification of elements of homotopy groups realizable by totally geodesic spheres in symmetric spaces of type I.
§11. Bott Periodicity and Its Relation with the Multidimensional Dirichlet Functional.
1. The explicit description of unitary Bott periodicity.
2. Unitary Bott periodicity follows from the properties of the two-dimensional extremals of the Dirichlet functional.
3. Orthogonal periodicity follows from the properties of the eight-dimensional extremals of the Dirichlet functional.
§12. Survey of Some Recent Results in Harmonic Mapping Theory.
References and Bibliography