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Multidimensional variational methods in the topology of extremals. (English. Russian original) Zbl 0576.49027
Russ. Math. Surv. 36, No. 6, 127-165 (1981); translation from Usp. Mat. Nauk 36, No. 6(222), 105-135 (1981).
This work gives a survey of methods, techniques and results in the theory of the volume functional on \(k\)-dimensional subsets of a Riemannian manifold \(M\) of dimension \(n\). Since the solutions of the Plateau problem are often singular on a Riemannian manifold and have an intricate topological structure, theorems on the regularity and existence of extremals are of special interest. In Section 2 the author compiles some statements about basic properties of extremals and explains the difference between local and global minimality. In Section 3, using the concept of stratified volume or “multivarifold”, he states conditions under which the minimum of the volume can be realized in a given homotopy, homology or bordism class of embeddings. The solution of the Plateau problem is stated here for the class of \(k\)-dimensional subsets which can be parametrized by a manifold \(V\). In Section 4 the author discusses methods for the construction of extremals. For example, for extremals in certain classes of \(k\)-dimensional subsets \(X\) of \(M\) and for suitable Morse functions \(f\) on \(M\), he gives estimates of the volume of the level sets \(X_ r\) of all \(x\in X\) with \(f(x)\leq r\). The section also contains some examples for minimal surfaces of nontrivial topological type.
In the last two sections the following problems are treated: Construction of minimal cones, determination of extremals which are invariant under the influence of a subgroup \(G\) of the isotropy group, construction of functions in \(\mathbb R^ n\) whose graphs are minimal surfaces, and finally the realization of nontrivial homotopy classes by harmonic mappings between spheres.

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
58E20 Harmonic maps, etc.
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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