The author’s present book attempts to give a concise account on the vast subject of topological variational problems.
In the first chapter, the main topological machinery is introduced, namely, singular and cellular (co)-homology theories, obstruction theory and fiber bundles. As problems of particular interest, existence of a retraction of a space onto a subspace homeomorphic to a sphere and the geometry of the Hopf bundle and the unit tangent bundle of spheres are also treated here.
In Chapter 2 the theory of exact Morse functions and a generalized Morse theory due to Novikov are developed. Next, under the title “Low-dimensional manifold”, the main topics include the study of groups of homeotopies and homeomorphisms of surfaces, the topology of 3-spheres and a detailed account of the Poincaré problem in dimension 4 (any homotopy sphere of dimension 4 is homeomorphic to \(S^4)\).
The core of the book is the last chapter on minimal surfaces which are introduced along with various points of view of the theory. Special emphasis is placed on stability, the classical Plateau principles and physical interpretation. This chapter contains a detailed account on the topological properties of minimal surfaces, especially: the topology of the boundary minimal cones, minimal surfaces invariant under a Lie group action, the Bernstein problem, bordisms and the higher-dimensional Plateau problem and the existence of minima in each homotopy class.
Finally, the last paragraph is devoted to the geometric properties of the extrema of the volume and Dirichlet functionals; it contains, among others, a lower bound for the volume of minimal surfaces, closed surfaces with nontrivial topology and minimal volume and nonexistence of local minima for the Dirichlet functional on maps of Riemannian manifolds into homogeneous spaces.
As the author’s previous books on the subject, this monograph is also among the few which treat difficult mathematics with amazing clarity and insight accompanied by highly effective visual illustrations. The author’s attempt to give a relatively short (and consequently, fairly concise) cross-section of the present-day knowledge on the topological variational problems proves to be successful; an English edition of this work is highly desirable.