The multidimensional Plateau problem in Riemannian manifolds

There always exists a soap film $X$ “spanning the hole” in a fixed closed wire contour $A$, and it turns out to be a minimal surface (i.e. any small perturbation increases its area). The mathematical solution of this two-dimensional Plateau problem was given by Douglas, Courant and Morrey. In dimensions greater than two, the multidimensional Plateau problem remained open. We shall consider the class of all $k$-dimensional films $X$ which have as a boundary a fixed $(k-1)$-dimensional submanifold $A$ such that each film $X$ admits a parametrization (i.e. it can be represented as the image of some manifold $W$ with boundary $A$ under a continuous function $f$ which is the identity on $A$). Is it possible to find a minimal film $X_0$ in this class? The solution of this problem, formulated in a new language, was obtained by using extraordinary homology and cohomology theories.
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