The present book is a new attempt to give a comprehensive, but concise, account to the vast subject of topological variational problems, with emphasis on the contemporary state of the theory of minimal surfaces and of one of the most important divisions of this theory - Plateau’s problem, i.e. the problem of the existence of a minimal surface with a prescribed boundary. Besides its intrinsic beauty and importance, the theory of minimal surfaces also has far reaching connections with the theory of differential equations, the theory of Lie groups and algebras, algebraic and differential topology, and the multidimensional calculus of variations.
The second named author is well known for his fundamental contributions to the development of the subject and also for his two previous books on it [Variational methods in topology (1982; Zbl 0526.58012), and Topological variational problems (1984)].
The first five chapters were written by A. T. Fomenko. In the long chapter 1, which is accessible to a wide public, he presents a historical survey of the development of the classical theory of two-dimensional minimal surfaces, from its beginnings in the 18th century until the early 20th century, including a description of the physical experiments which led to the formulation of the classical Plateau problem, its solution by Douglas and Rado, and the latest developments in relation with other disciplines, as the modern theory of singularities. This is followed by a short chapter 2 which presents the basic facts from homology theory, necessary in later exposition. Chapter 3 surveys the contemporary state of the theory of multidimensional minimal surfaces in Riemannian manifolds. It treats a great variety of results, concerning the locally minimal surfaces in arbitrary codimensions, the global minimality of complex submanifolds, the author’s solution of Plateau’s problem in the class of homological surfaces with a fixed homological boundary (and examples which prove the insufficiency of this class), the solution of Plateau’s problem in classes of currents and varifolds, and the theory of minimal cones with its relations to the equivariant Plateau problem.
The classical multidimensional Plateau problem in then extended in chapter 4 by reformulating it in the context of singular bordisms; its solution by the author is given in terms of two important notions: stratified surfaces and the stratified volume of a surface. The problem is then further extended and solved in terms of generalized (co)homology theories. Methods for constructing global minimal surfaces are derived. The next chapter 5 deals with the study of harmonic surfaces (extremals of the Dirichlet functional) and, in particular, of the totally geodesic surfaces in symmetric spaces, for which the realizability as nontrivial cycles is proven. Some open problems are stated.
The remaining five chapters, intended for specialists, were written by the first named author and are devoted to his special direction of research: the homotopical variant of Plateau’s problem. In chapter 6 the author introduces the notion of a multivarifold, as a functional analogue of a geometric stratified surface, and which is used in his solution of Plateau’s problem in a homotopy class. The topology of the space of multivarifolds is studied in chapter 7. Then, in chapter 8, parametrizations and parametrized multivarifolds are defined, and their metric and topological properties are derived. Chapter 9 contains the proof of a ”parametrized” version of the Federer-Fleming deformation theorem and of some related isoperimetric inequalities. The homotopical variant of Plateau’s problem is then reformulated as a very general variational problem in terms of parametrizations; its solution is given in terms of minimal parametrizations and parametrized multivarifolds. In the last chapter 10, criteria are derived in the general context of the functional spaces of currents.
The text is illustrated by a large number of pictures drawn by A. T. Fomenko with the same talent and imagination as in previous books of his and of his colleagues. The book certainly deserves a wider distribution through an English translation.