View source for Plateau problem, multi-dimensional

A term denoting a number of problems related to research on extremals and global minima of functionals in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728501.png" />-dimensional volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728502.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728503.png" />-dimensional generalized surfaces that are imbedded in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728504.png" />-dimensional Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728505.png" /> and that satisfy certain boundary conditions.

In the history of this variational problem (see [[Plateau problem|Plateau problem]]) one may distinguish several periods, characterized by different approaches to the concepts of a  "surface" , a  "boundary"  and  "minimization" , and, correspondingly, to methods of obtaining the minimal solution. The multi-dimensional Plateau problem is formulated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728506.png" /> be a fixed closed smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728507.png" />-dimensional submanifold in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728508.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p0728509.png" /> be the class of all films (surfaces) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285010.png" /> having <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285011.png" /> as boundary. Here, each film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285012.png" /> allows a continuous parametrization (it can be represented as the transform of some manifold with boundary), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285014.png" /> is a certain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285015.png" />-dimensional manifold with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285016.png" /> homeomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285017.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285018.png" /> is a continuous mapping that coincides with a given homeomorphism at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285019.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285020.png" />. The question is whether one can find in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285021.png" /> a film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285022.png" /> that would in some reasonable sense be minimal, i.e. such that its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285023.png" />-dimensional volume is less than those of other films <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285024.png" /> of the same class. The transfer of classical  "two-dimensional"  methods to the multi-dimensional case encounters serious difficulties. E.g., the classical formulation of the multi-dimensional Plateau problem was set aside for some time and the problem was formulated in other (homological) terms. If one discards the concept of a manifold-film with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285025.png" /> and extensively generalizes the concepts of a film and its boundary, while weakening the link between the two (in particular, if one considers non-parametrized films), and if one discards the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285026.png" />, then the multi-dimensional problem can be formulated in the language of the ordinary integer homology groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285027.png" />: Find the minimal film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285028.png" /> that annihilates the fundamental cycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285029.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285030.png" /> (on the assumption that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285031.png" /> can be oriented), i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285033.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285034.png" /> is the homomorphism induced by the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285035.png" />. To solve the multi-dimensional problem in this new and extended formulation, a geometrical approach has been developed [[#References|[1]]], [[#References|[2]]], in which one minimizes a function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285036.png" />-dimensional Hausdorff measure (volume) defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285037.png" />-dimensional measurable compact sets (surfaces) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285038.png" />, and a theory has been developed [[#References|[3]]], [[#References|[4]]] of integral currents and varifolds with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285039.png" />-rectifiable subsets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285040.png" /> as supports. In both directions fundamental theorems on the existence of minimal surfaces with a given boundary have been proved (E.R. Reifenberg, C.B. Morrey, H. Federer, W. Fleming, F.J. Almgren, E. de Giorgi, R. Harvey, H.B. Lawson, J. Simons, E. Giusti, and others). For a review on the tremendous literature on this question, see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]. In particular, in the well-known work of Reifenberg the multi-dimensional Plateau problem has been solved in terms of spectral homology (Čech homology). It was also proved that a minimal surface, spanned over a  "multi-dimensional"  contour, is a manifold of corresponding smoothness class almost-everywhere, except at, possibly, a set of singular points of measure zero. Famous theorems on the existence of minimal surfaces in terms of minimal currents, minimal varifolds with a fixed boundary or without boundary, were proved by Federer, Fleming and Almgren [[#References|[3]]], [[#References|[4]]]. As in the previous case, the minimal surfaces turned out to be manifolds outside, possibly, a set of singular points of measure zero. Later the equivariant Plateau problem was solved by W.Y. Hsiang and Lawson [[#References|[16]]]. More precisely, the existence of a minimal surface in a Euclidean space having the same symmetry groups on the multi-dimensional boundary  "contour"  has been proved. The transition of this theorem to arbitrary Riemannian manifolds was completed by J.E. Brothers [[#References|[17]]]. The complex version of Plateau's problem was obtained by Harvey and Lawson . In particular, existence conditions for complex minimal films with a given boundary were discovered. Later, the  "Lagrange variant"  of the multi-dimensional Plateau problem was obtained by Harvey and Lawson [[#References|[19]]]. As a result, minimality conditions for Lagrange submanifolds in the symplectic complex linear space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285041.png" /> were obtained. Deep results and existence theorems for minimal surfaces were obtained by S.S.-T. Yau [[#References|[20]]]. He revealed the link between the existence of complex minimal surfaces and Kohn–Rossi cohomology. Using the theory of minimal surfaces, clear results were obtained by W.H. Meeks and Yau on the theory of three-dimensional manifolds, [[#References|[21]]]. Note that if one has a theorem on the existence of a minimal solution in the homology class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285042.png" />, one can as before still say nothing on the existence of a minimal solution in the class of all films that are continuous transforms of manifolds with boundaries, i.e. that allow a parametrization. The fact is that if a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285043.png" /> is homologous to zero (as a cycle) in a film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285044.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285045.png" /> does not necessarily have a representation in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285046.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285047.png" /> is some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285048.png" />-dimensional manifold with boundary.

In [[#References|[5]]], [[#References|[6]]] a solution to a variant of the multi-dimensional Plateau problem was given in terms of spectral bordism. The spectral bordism groups are defined, for any compact space, using a Čech process analogous to the definition of spectral (Čech) homology. This process allows one to extend ordinary bordism groups of polyhedral classes and cell complexes to the wider class of compacta (e.g., in Riemannian manifolds). An element of a spectral bordism group can be represented by a sequence of manifolds, connected by mappings. For finite cell complexes this element is represented by one manifold (ordinary bordism). If a topological space is, e.g., a finite cell complex, then its spectral bordism group coincides with the ordinary singular bordisms. It was found that the classical problem has an equivalent formulation in the language of bordisms (cf. [[Bordism|Bordism]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285049.png" /> be a compact oriented closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285050.png" />-dimensional manifold and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285051.png" /> be a continuous mapping; the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285052.png" /> is called a singular bordism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285053.png" />. Two bordisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285055.png" /> are said to be equivalent if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285056.png" />-dimensional oriented manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285057.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285058.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285059.png" /> denotes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285060.png" /> with the opposite orientation) and a continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285061.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285063.png" />. The bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285064.png" /> is equivalent to zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285066.png" />. The equivalence classes of singular bordisms form an Abelian group, which after stabilization forms one of the generalized homology theories (bordism theory). The multi-dimensional Plateau problem is formulated (in this language) as follows: a) Can one find an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285067.png" /> with least volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285068.png" /> among all films <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285070.png" />, having the property that the singular bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285071.png" /> is equivalent to zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285072.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285073.png" /> is the imbedding? b) Can one find a bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285074.png" /> among all singular bordisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285075.png" /> equivalent to a given bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285076.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285077.png" />) such that the volume of the film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285078.png" /> is minimal?

The classical multi-dimensional Plateau problem differs considerably from the homological variant.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072850a.gif" />

Figure: p072850a

Fig. a shows the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285079.png" /> and the film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285080.png" /> that tends to occupy a position in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285081.png" /> corresponding to minimal area. At a certain instant, the film links up and collapses, and instead of the two-dimensional tube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285082.png" /> one gets a one-dimensional segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285083.png" />. In the two-dimensional case, the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285084.png" /> may be mapped continuously into a two-dimensional disc glued to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285085.png" />. In the multi-dimensional case, this effect of zones with fewer dimensions occurring in a minimal film is present to an even greater extent, and whereas all such parts <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285087.png" />, in the two-dimensional case can be mapped without loss of the parametrizing properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285088.png" /> into a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285089.png" />-dimensional (two-dimensional) part of this film, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285090.png" /> these zones of fewer dimensions, in general, cannot be eliminated (if one wishes to retain the topological property of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285091.png" /> of annihilating the bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285092.png" />). For the same reasons, the zones of fewer dimensions cannot be discarded, since a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285093.png" />-dimensional part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285094.png" /> of a film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285095.png" /> need not have a continuous parametrization, and thus, generally speaking, need not annihilate the bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285096.png" />. This shows that it is necessary to introduce the stratified volume of the film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285097.png" />, composed of the volumes of all zones <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285098.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p07285099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850100.png" />. A theorem representing a solution to a variant of the multi-dimensional Plateau problem in terms of spectral bordism is as follows [[#References|[5]]], [[#References|[6]]]: There exists a globally-minimal surface that minimizes the stratified volume.

Consequence: For any fixed oriented smooth closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850101.png" />-dimensional submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850102.png" /> in a Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850103.png" /> (in the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850104.png" />), there exists a globally-minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850105.png" /> that annihilates the spectral bordism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850106.png" />. If the minimal film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850107.png" /> is a finite cell complex, it is representable in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850109.png" /> is some manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850110.png" /> is a mapping that is a homeomorphism from the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850111.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850112.png" /> onto the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850113.png" />, [[#References|[5]]], . Also, the film <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850114.png" /> is minimal in each of its dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850115.png" />; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850116.png" /> is the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850117.png" /> having dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850118.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850119.png" /> contains a subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850120.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850121.png" />-dimensional volume zero, while the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850122.png" /> is an open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850123.png" />-dimensional everywhere-dense analytic submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850124.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850125.png" /> is the set of singular points in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850126.png" />.

This result is a particular case of a general theorem on the existence and almost-everywhere regularity of a globally-minimal surface, which has been proved [[#References|[5]]], [[#References|[6]]],

for any generalized (co)homology theory and for any set of boundary conditions. Also, such a surface exists in each stable homotopy class. The following is an example of a variational problem formulated and solved in cohomology terms. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850127.png" /> be a stably non-trivial vector bundle on a compact Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850128.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850129.png" /> be the class of all surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850130.png" /> such that the restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850131.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850132.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850133.png" /> is stably non-trivial (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850134.png" /> is the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850135.png" />). Then there always exists a globally-minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850136.png" /> having least volume in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850137.png" />. The general existence theorem can be formulated and proved also in the language of integral currents, for which one introduces filtered currents consisting of currents with various dimensions. In this way a solution to the multi-dimensional Plateau problem was obtained in homotopy classes of multi-varifolds [[#References|[14]]].

In the circle of problems related to the multi-dimensional Plateau problem one may distinguish research on particular analytic and topological features of globally-minimal surfaces. For example, there is the current problem of representing particular surfaces in Riemannian spaces. For example, it is known [[#References|[3]]] that complex-algebraic subvarieties in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850138.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850139.png" /> are globally-minimal surfaces. One of the results has an explicit complex character. In the case of real subvarieties, for a long time there were no methods for detecting particular globally-minimal surfaces. The first result in this field [[#References|[6]]], which incorporated the topology, was a method that enabled one to show that each compact Riemannian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850140.png" /> can be put into correspondence with a universal function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850141.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850142.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850143.png" /> is an integer, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850144.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850145.png" /> is a globally-minimal surface that realizes a non-trivial (co)cycle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850146.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850147.png" /> for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850148.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850149.png" /> is a homogeneous space, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850150.png" /> is independent of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850151.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850152.png" /> is calculated in explicit form and gives a general lower bound to the volumes of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850153.png" />-dimensional (co)cycles in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850154.png" />. This bound cannot be improved in the general case, i.e. there exist infinite series of globally-minimal films <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850155.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850156.png" />. A result for symmetric spaces ([[#References|[6]]], , [[#References|[15]]]) is a complete description of all surfaces for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850157.png" />. Further methods have been devised [[#References|[11]]], [[#References|[12]]], [[#References|[14]]], [[#References|[15]]] for obtaining particular globally-minimal surfaces.

There are various problems in variational calculus, topology, algebraic geometry, and complex analysis that give rise to the following situation: One is given a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850158.png" /> and an exhaustion of it by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850159.png" />-dimensional regions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850160.png" /> that expand as the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850161.png" /> increases; in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850162.png" />, there is a definite globally-minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850163.png" />; the question is raised of the rate at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850164.png" /> increases, considered as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850165.png" />. This question arises for example in the calculation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850166.png" />, in the problem of the structure of bases in spaces of entire functions, in theorems of Stoll type [[#References|[11]]], etc. It has been found [[#References|[6]]], [[#References|[15]]] that there exists a universal exact effectively-computable lower bound to the rate of increase of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850167.png" />, which implies as particular cases explicit formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850168.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850169.png" /> is a globally-minimal surface. For example, the volume of such a surface enclosed in a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850170.png" /> and passing through the centre of the sphere (and having its boundary at the boundary of the sphere) is always at least that of a standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850171.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850172.png" /> (a planar section) passing through the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850173.png" /> , [[#References|[15]]].

A particular line of research is represented by the multi-dimensional Plateau problem of codimension one: One considers globally-minimal surfaces of codimension 1 in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850174.png" />. For example, Bernstein's problem (S.N. Bernshtein) has been solved [[#References|[7]]]: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850176.png" /> be a smooth complete locally-minimal submanifold in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850177.png" /> allowing of one-to-one projection onto a certain hyperplane, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850178.png" /> is given by the graph of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850179.png" /> defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850180.png" />; is it true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850181.png" /> is a linear function? The answer is positive for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850182.png" /> [[#References|[8]]]. The minimality of such hypersurfaces is closely related to the minimality of cones in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850183.png" />: The existence of a locally-minimal surface implies the existence of a minimal cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850184.png" />, i.e. of a surface composed of points on radii running from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850185.png" /> to points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850186.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850187.png" /> is a locally-minimal surface in the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850188.png" />. It has been established [[#References|[8]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850189.png" /> is a closed locally-minimal submanifold (i.e. one that annihilates the Euler operator) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850190.png" /> that is not the equator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850191.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850192.png" /> the cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850193.png" /> with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850194.png" /> and vertex at the centre of the sphere does not minimize the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850195.png" />-dimensional volume <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850196.png" /> (for a fixed boundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850197.png" />), i.e. there exists a variation (with support localized around the centre of the sphere) that reduces the volume of the cone. This implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850198.png" /> is a linear function for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850199.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850200.png" /> the answer is negative: There exist [[#References|[7]]] locally- and even globally-minimal surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850201.png" /> defined as graphs of non-linear functions. The construction can be performed explicitly; it is then found that cones specified in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850202.png" /> by the equation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

are globally-minimal surfaces with a fixed boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850205.png" />. These cones represent a particular case of cones of more general form that are globally-minimal surfaces [[#References|[10]]].

There is a new line of research on the multi-dimensional Plateau problem, namely that of equivariant multi-dimensional Plateau problems. Among the globally-minimal surfaces, one naturally distinguishes the class of films that are transformed into themselves under the action of a certain symmetry group [[#References|[9]]], [[#References|[10]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850206.png" /> be a compact connected Lie group acting smoothly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850207.png" /> by isometries and stratifying it into orbits <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850209.png" />. Then to find the globally-minimal surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850210.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850211.png" /> that are invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850212.png" /> it is sufficient to transfer to the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850213.png" /> and endow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850214.png" /> with a Riemannian metric of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850215.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850216.png" /> and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850217.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850218.png" /> denotes the dimension of an orbit in general position in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850219.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850220.png" /> is the natural projection metric arising in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850221.png" /> under the isometric action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850222.png" />. To find globally-minimal surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850223.png" /> invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850224.png" /> it is sufficient to describe such surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850225.png" /> endowed with the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850226.png" /> [[#References|[9]]], so one gets a reduction of the multi-dimensional Plateau problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850227.png" /> to the same problem in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850228.png" /> in fewer dimensions. This method has provided a number of particular globally-minimal surfaces having large symmetry groups .

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/p072850b.gif" />

Figure: p072850b

In particular,  "Simons cones" , defined by (*), are represented by the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850229.png" /> (Fig. b) on the two-dimensional quotient space

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850230.png" /></td> </tr></table>

endowed with the metric

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850231.png" /></td> </tr></table>

and representing the first quadrant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850232.png" /> in the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850233.png" /> [[#References|[10]]]. To find a globally-minimal surface with boundary

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850234.png" /></td> </tr></table>

it is sufficient to find geodesics running from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850235.png" /> to the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850236.png" /> and having minimum length. Fig. bshows a pencil of geodesics running from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850237.png" />; this pencil can be understood as a pencil of light rays propagating from a source <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850238.png" /> in a transparent medium filling <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850239.png" /> with refraction index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850240.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850241.png" />, in addition to the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850242.png" /> there exists a further minimal solution of smaller length, which is represented by the geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850243.png" />; this means that this Simons cone is not a globally-minimal surface. As <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850244.png" /> increases, the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850245.png" /> tends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850246.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850247.png" /> there exists a unique geodesic joining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850248.png" /> to the boundary of the quadrant, i.e. this Simons cone is a globally-minimal surface [[#References|[10]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Morrey,   "Multiple integrals in the calculus of variations" , Springer  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Reifenberg,   "Solution of Plateau's problem for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850249.png" />-dimensional surfaces of varying topological type"  ''Acta Math.'' , '''104''' :  1/2  (1960)  pp. 1–92</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Federer,   "Geometric measure theory" , Springer  (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.J. Almgren,   "Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure"  ''Ann. of Math. (2)'' , '''87''' :  2  (1968)  pp. 321–391</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.T. Fomenko,   "The multidimensional Plateau problem in Riemannian manifolds"  ''Math. USSR Sb.'' , '''18''' :  3  (1972)  pp. 487–527  ''Mat. Sb.'' , '''89''' :  3  (1972)  pp. 475–519</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  A.T. Fomenko,   "Minimal compacta in Riemannian manifolds and Reifenberg's conjecture"  ''Math. USSR Izv.'' , '''6''' :  5  (1972)  pp. 1037–1066  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''36''' :  5  (1972)  pp. 1049–1079</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. Bombieri,   E. de Giorgi,   E. Giusti,   "Minimal cones and the Bernstein problem"  ''Invent. Math.'' , '''7''' :  3  (1969)  pp. 243–268</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Simons,   "Minimal varieties in Riemannian manifolds"  ''Ann. of Math.'' , '''88''' :  1  (1968)  pp. 62–105</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  H. Lawson,   "The equivalent Plateau problem and interior regularity"  ''Trans. Amer. Math. Soc.'' , '''173''' :  446  (1972)  pp. 231–249</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  H. Lawson,   J. Simons,   "On stable currents and their application to global problems in real and complex geometry"  ''Ann. of Math.'' , '''98''' :  3  (1973)  pp. 427–450</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  A.T. Fomenko,   "Bott periodicity from the point of view of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850250.png" />-dimensional Dirichlet functional"  ''Math. USSR Izv.'' , '''5''' :  3  (1971)  pp. 681–695  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35''' :  3  (1971)  pp. 667–681</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  Dao Chong Tkhi,   "Multivarifolds and classical multidimensional Plateau problems"  ''Math. USSR Izv.'' , '''17'''  (1981)  pp. 271–298  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''44''' :  5  (1980)  pp. 1031–1065</TD></TR><TR><TD valign="top">[13a]</TD> <TD valign="top">  A.T. Fomenko,   "Multidimensional Plateau problem in Riemannian manifolds and extraordinary homology and cohomology theories I" , ''Proc. Sem. Vektor. Tenzor. Anal.'' , '''17'''  (1974)  pp. 3–176  (In Russian)</TD></TR><TR><TD valign="top">[13b]</TD> <TD valign="top">  A.T. Fomenko,   "Multidimensional Plateau problem in Riemannian manifolds and extraordinary homology and cohomology theories II" , ''Proc. Sem. Vektor. Tenzor. Anal.'' , '''18'''  (1978)  pp. 4–93  (In Russian)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  Dao Chong Tkhi,   "On the stability of the homology of compact Riemannian manifolds"  ''Math. USSR Izv.'' , '''12'''  (1978)  pp. 463–468  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''42''' :  3  (1978)  pp. 500–505</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  A.T. Fomenko,   "Multi-dimensional variational methods in the topology of extremals"  ''Russian Math. Surveys'' , '''36''' :  6  (1981)  pp. 127–165  ''Uspekhi Mat. Nauk'' , '''36''' :  6  (1981)  pp. 125–169</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  W.Y. Hsiang,   H.B. Lawson,   "Minimal submanifolds of low cohomogenety"  ''J. Diff. Geom.'' , '''5''' :  1  (1971)  pp. 1–38</TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top">  J.E. Brothers,   "Invariance of solutions to invariant parametric variational problems"  ''Trans. Amer. Math. Soc.'' , '''262''' :  1  (1980)  pp. 159–180</TD></TR><TR><TD valign="top">[18a]</TD> <TD valign="top">  R. Harvey,   H.B. Lawson,   "On boundaries of complex analytic varieties I"  ''Ann. of Math.'' , '''102'''  (1975)  pp. 233–290</TD></TR><TR><TD valign="top">[18b]</TD> <TD valign="top">  R. Harvey,   H.B. Lawson,   "On boundaries of complex analytic varieties II"  ''Ann. of Math.'' , '''106'''  (1977)  pp. 213–238</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top">  R. Harvey,   H.B. Lawson,   "Calibrated geometries"  ''Acta Math.'' , '''148'''  (1982)  pp. 47–157</TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top">  S.S.-T. Yau,   "Kohn–Rossi cohomology and its application to the complex Plateau problem"  ''Ann. of Math.'' , '''113''' :  1  (1981)  pp. 67–110</TD></TR><TR><TD valign="top">[21]</TD> <TD valign="top">  W.H. Meeks,   S.S.-T. Yau,   "The classical Plateau problem and the topology of three-dimensional manifolds. The embedding of the solution given by Dehn's lemma"  ''Topology'' , '''21''' :  4  (1982)  pp. 409–442</TD></TR><TR><TD valign="top">[22]</TD> <TD valign="top">  A.T. Fomenko,   "Plateau problem" , '''1–2''' , Gordon &amp; Breach  (1990)  (Translated from Russian)</TD></TR></table>



====Comments====
For minimal surfaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072850/p072850251.png" /> see also [[#References|[a4]]], [[#References|[a5]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.T. Fomenko,   "Variational principles in topology - Multidimensional minimal surface theory" , Kluwer  (1990)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Giusti,   "Minimal surfaces and functions of bounded variation" , Birkhäuser  (1984)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.T. Fomenko,   Dao Chong Tkhi,   "Minimal surfaces and Plateau's problem" , Amer. Math. Soc.  (Forthcoming)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G. Stolzenberg,   "Volumes, limits, and extensions of analytic varieties" , Springer  (1966)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.M. Chirka,   "Complex analytic sets" , Kluwer  (1989)  pp. §19  (Translated from Russian)</TD></TR></table>