The Plateau problem, Part 2. Present state of the theory

Cover......Page p0001.djvu
Volume 1: II Present State of the Theory......Page p0002.djvu
THE PLATEAU PROBLEM:The Present State of the Theory. Part II......Page p0003.djvu
©......Page p0004.djvu
CONTENTS......Page p0005.djvu
List of Contents of Part I......Page p0008.djvu
Preface......Page p0011.djvu
1. Two-dimensional minimal surfaces in Euclidean space and in Riemannian manifolds.......Page p0013.djvu
2. The multi-dimensional Riemannian volume functional and second fundamental form on a submanifold.......Page p0021.djvu
3. Multidimensional locally minimal surfaces......Page p0028.djvu
4. The global minimality of complex submanifolds......Page p0030.djvu
5. The complex Plateau problem......Page p0032.djvu
6. On various approaches to the concepts of surface and boundary of a surface......Page p0033.djvu
7. The homology boundary of a surface and the role of the coefficient group......Page p0036.djvu
8. Surprising examples of physical stable minimal surfaces, that nevertheless retract onto their boundaries......Page p0040.djvu
9. When does a soap film spanning a frame not contain closed soap-bubbles?......Page p0053.djvu
§2 Integral Currents\\r......Page p0060.djvu
1. De Rham currents. Basic notions.......Page p0063.djvu
2. Rectifiable currents and flat chains......Page p0066.djvu
3. Normal and integral currents......Page p0067.djvu
4. Various formulations of the minimal current existence theorem.......Page p0069.djvu
5. Varifolds and minimal surfaces.......Page p0071.djvu
6. The interior regularity for minimal surfaces and the structure of their singular points.......Page p0077.djvu
7. Regularity almost everywhere for the supports of ellipticintegrand-minimizing k-currents and k-varifolds.......Page p0079.djvu
8. The interior regularity for volume-minimizing hypersurfaces and the existence of minimal cones of codimension one.......Page p0081.djvu
9. Dimension estimates for the set of singular points of a minimal surface.......Page p0083.djvu
10. Other problems of minimal surface regularity.......Page p0086.djvu
1. Minimal cones associated with singular points of minimal surfaces.......Page p0087.djvu
2. Multidimensional minimal cones.......Page p0090.djvu
3. Minimal surfaces invariant with respect to the action of Lie groups.......Page p0100.djvu
4. The Fermat principle, minimal cones, and light rays.......Page p0105.djvu
5. S.N. Bernstein\'s problem.......Page p0116.djvu
1. Bordant manifolds and the multidimensional Plateau problem.......Page p0120.djvu
2. The properties of bordant manifold classes......Page p0127.djvu
3. The statement of the existence theorem for globally minimal surfaces in the spectral bordant manifold class.......Page p0135.djvu
1. The definition of generalized homology and cohomology.......Page p0138.djvu
2. The coboundary and boundary of a pair of spaces (X,A)......Page p0141.djvu
3. Surface variational classes.......Page p0142.djvu
4. The general existence theorem for globally minimal surfaces in an arbitrary class determined by a generalized spectral homology or cohomology theory.......Page p0144.djvu
5. A short sketch of the proof of Theorem 1.......Page p0147.djvu
1 The functional multivarifold language.......Page p0154.djvu
2. Multivarifolds and variational problems in classes of surfaces of fIxed topological type.......Page p0162.djvu
3. Minimization problems for generalized integrands in the parametrization and parametrized multivarifold classes.......Page p0166.djvu
4. Criteria for the global minimality of surfaces and currents.......Page p0170.djvu
§7 Cases where a Solution of the Dirichlet Problem for the Equation of Minimal Surfaces of High Codimensions does not Exist......Page p0177.djvu
§8 Example of a Smooth, Closed, Unknotted Curve inR3, Bounding Only Minimal Surfaces of Large Genus.......Page p0187.djvu
1. The universal lower estimate of the volumes of topologically non-trivial minimal surfaces.......Page p0194.djvu
2. The coefficient of deformation of a vector field.......Page p0197.djvu
3. Surfaces of non-trivial topological type and of least volume.......Page p0198.djvu
4 On the minimal volume of surfaces passing through the centre of a symmetric convex domain in Euclidean space.......Page p0209.djvu
1. Totally geodesic submanifolds in Lie groups.......Page p0213.djvu
3. When does a totally geodesic submanifold realize a non-trivial cycle?......Page p0215.djvu
4. The classification theorem for totally geodesic submanifolds realizing non-trivial cycles in symmetric spaces.......Page p0219.djvu
5. The classification of cocycles realizable by totally geodesic spheres in compact Lie groups.......Page p0220.djvu
6. The classification of elements of homotopy groups realizable by totally geodesic spheres in symmetric spaces of type I.......Page p0223.djvu
1. The explicit description of unitary Bott periodicity.......Page p0226.djvu
2. Unitary Bott periodicity follows from the properties of the two-dimensional extremals of the Dirichlet functional.......Page p0229.djvu
3. Orthogonal periodicity follows from the properties of the eight-dimensional extremals of the Dirichlet functional.......Page p0231.djvu
§12 Survey of Some Recent Results in Harmonic Mapping Theory.......Page p0233.djvu
References and Bibliography......Page p0246.djvu
INDEX......Page p0261.djvu
Back......Page p0264.djvu