Plateau problem, multi-dimensional
A term denoting a number of problems related to research on extremals and global minima of functionals in the -dimensional volume
defined on
-dimensional generalized surfaces that are imbedded in an
-dimensional Riemannian space
and that satisfy certain boundary conditions.
In the history of this variational problem (see 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 be a fixed closed smooth
-dimensional submanifold in a Riemannian space
and let
be the class of all films (surfaces)
having
as boundary. Here, each film
allows a continuous parametrization (it can be represented as the transform of some manifold with boundary), i.e.
, where
is a certain
-dimensional manifold with boundary
homeomorphic to
, while
is a continuous mapping that coincides with a given homeomorphism at
, i.e.
. The question is whether one can find in the class
a film
that would in some reasonable sense be minimal, i.e. such that its
-dimensional volume is less than those of other films
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
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
, then the multi-dimensional problem can be formulated in the language of the ordinary integer homology groups
: Find the minimal film
that annihilates the fundamental cycle
of a manifold
(on the assumption that
can be oriented), i.e.
,
, where
is the homomorphism induced by the imbedding
. To solve the multi-dimensional problem in this new and extended formulation, a geometrical approach has been developed [1], [2], in which one minimizes a function of the
-dimensional Hausdorff measure (volume) defined on
-dimensional measurable compact sets (surfaces) in
, and a theory has been developed [3], [4] of integral currents and varifolds with
-rectifiable subsets in
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 [1], [3], [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 [3], [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 [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 [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 [19]. As a result, minimality conditions for Lagrange submanifolds in the symplectic complex linear space
were obtained. Deep results and existence theorems for minimal surfaces were obtained by S.S.-T. Yau [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, [21]. Note that if one has a theorem on the existence of a minimal solution in the homology class
, 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
is homologous to zero (as a cycle) in a film
, then
does not necessarily have a representation in the form
, where
is some
-dimensional manifold with boundary.
In [5], [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). Let be a compact oriented closed
-dimensional manifold and let
be a continuous mapping; the pair
is called a singular bordism of
. Two bordisms
and
are said to be equivalent if there exists a
-dimensional oriented manifold
with boundary
(where
denotes
with the opposite orientation) and a continuous mapping
such that
,
. The bordism
is equivalent to zero if
,
. 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
with least volume
among all films
,
, having the property that the singular bordism
is equivalent to zero in
, where
is the imbedding? b) Can one find a bordism
among all singular bordisms
equivalent to a given bordism
(where
) such that the volume of the film
is minimal?
The classical multi-dimensional Plateau problem differs considerably from the homological variant.
Figure: p072850a
Fig. a shows the contour and the film
that tends to occupy a position in
corresponding to minimal area. At a certain instant, the film links up and collapses, and instead of the two-dimensional tube
one gets a one-dimensional segment
. In the two-dimensional case, the segment
may be mapped continuously into a two-dimensional disc glued to
. 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
,
, in the two-dimensional case can be mapped without loss of the parametrizing properties of
into a
-dimensional (two-dimensional) part of this film, for
these zones of fewer dimensions, in general, cannot be eliminated (if one wishes to retain the topological property of
of annihilating the bordism
). For the same reasons, the zones of fewer dimensions cannot be discarded, since a
-dimensional part
of a film
need not have a continuous parametrization, and thus, generally speaking, need not annihilate the bordism
. This shows that it is necessary to introduce the stratified volume of the film
, composed of the volumes of all zones
, i.e.
,
. A theorem representing a solution to a variant of the multi-dimensional Plateau problem in terms of spectral bordism is as follows [5], [6]: There exists a globally-minimal surface that minimizes the stratified volume.
Consequence: For any fixed oriented smooth closed -dimensional submanifold
in a Riemannian space
(in the case where
), there exists a globally-minimal surface
that annihilates the spectral bordism
. If the minimal film
is a finite cell complex, it is representable in the form
, where
is some manifold and
is a mapping that is a homeomorphism from the boundary
of
onto the manifold
, [5], . Also, the film
is minimal in each of its dimensions
; if
is the part of
having dimension
, then
contains a subset
of
-dimensional volume zero, while the complement
is an open
-dimensional everywhere-dense analytic submanifold in
. Here
is the set of singular points in dimension
.
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 [5], [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 be a stably non-trivial vector bundle on a compact Riemannian space
; let
be the class of all surfaces
such that the restriction
of
to
is stably non-trivial (i.e.,
is the support of
). Then there always exists a globally-minimal surface
having least volume in the class
. 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 [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 [3] that complex-algebraic subvarieties in and
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 [6], which incorporated the topology, was a method that enabled one to show that each compact Riemannian space
can be put into correspondence with a universal function
, where
and
is an integer,
. If
is a globally-minimal surface that realizes a non-trivial (co)cycle in
, then
for any point
. If
is a homogeneous space, then
is independent of the point
. The function
is calculated in explicit form and gives a general lower bound to the volumes of all
-dimensional (co)cycles in
. This bound cannot be improved in the general case, i.e. there exist infinite series of globally-minimal films
for which
. A result for symmetric spaces ([6], , [15]) is a complete description of all surfaces for which
. Further methods have been devised [11], [12], [14], [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 and an exhaustion of it by
-dimensional regions
that expand as the parameter
increases; in
, there is a definite globally-minimal surface
; the question is raised of the rate at which
increases, considered as a function of
. This question arises for example in the calculation of
, in the problem of the structure of bases in spaces of entire functions, in theorems of Stoll type [11], etc. It has been found [6], [15] that there exists a universal exact effectively-computable lower bound to the rate of increase of
, which implies as particular cases explicit formulas for
, where
is a globally-minimal surface. For example, the volume of such a surface enclosed in a sphere
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
-dimensional sphere
(a planar section) passing through the centre of
, [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 . For example, Bernstein's problem (S.N. Bernshtein) has been solved [7]: Let
be a smooth complete locally-minimal submanifold in
allowing of one-to-one projection onto a certain hyperplane, i.e.
is given by the graph of a function
defined in
; is it true that
is a linear function? The answer is positive for
[8]. The minimality of such hypersurfaces is closely related to the minimality of cones in
: The existence of a locally-minimal surface implies the existence of a minimal cone
, i.e. of a surface composed of points on radii running from a point
to points
, where
is a locally-minimal surface in the sphere
. It has been established [8] that if
is a closed locally-minimal submanifold (i.e. one that annihilates the Euler operator) in
that is not the equator
, then for
the cone
with base
and vertex at the centre of the sphere does not minimize the
-dimensional volume
(for a fixed boundary in
), 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
is a linear function for
. For
the answer is negative: There exist [7] locally- and even globally-minimal surfaces
defined as graphs of non-linear functions. The construction can be performed explicitly; it is then found that cones specified in
by the equation
![]() | (*) |
are globally-minimal surfaces with a fixed boundary ,
. These cones represent a particular case of cones of more general form that are globally-minimal surfaces [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 [9], [10]. Let be a compact connected Lie group acting smoothly on
by isometries and stratifying it into orbits
,
. Then to find the globally-minimal surfaces
in
that are invariant with respect to
it is sufficient to transfer to the quotient space
and endow
with a Riemannian metric of the form
![]() |
where and
![]() |
where denotes the dimension of an orbit in general position in
and
is the natural projection metric arising in
under the isometric action of
. To find globally-minimal surfaces in
invariant with respect to
it is sufficient to describe such surfaces in
endowed with the metric
[9], so one gets a reduction of the multi-dimensional Plateau problem in
to the same problem in
in fewer dimensions. This method has provided a number of particular globally-minimal surfaces having large symmetry groups .
Figure: p072850b
In particular, "Simons cones" , defined by (*), are represented by the line (Fig. b) on the two-dimensional quotient space
![]() |
endowed with the metric
![]() |
and representing the first quadrant in the plane
[10]. To find a globally-minimal surface with boundary
![]() |
it is sufficient to find geodesics running from to the boundary of
and having minimum length. Fig. bshows a pencil of geodesics running from
; this pencil can be understood as a pencil of light rays propagating from a source
in a transparent medium filling
with refraction index
. For
, in addition to the surface
there exists a further minimal solution of smaller length, which is represented by the geodesic
; this means that this Simons cone is not a globally-minimal surface. As
increases, the point
tends to
, and for
there exists a unique geodesic joining
to the boundary of the quadrant, i.e. this Simons cone is a globally-minimal surface [10].
References
[1] | C. Morrey, "Multiple integrals in the calculus of variations" , Springer (1966) |
[2] | E. Reifenberg, "Solution of Plateau's problem for ![]() |
[3] | H. Federer, "Geometric measure theory" , Springer (1969) |
[4] | 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 |
[5] | 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 |
[6] | 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 |
[7] | E. Bombieri, E. de Giorgi, E. Giusti, "Minimal cones and the Bernstein problem" Invent. Math. , 7 : 3 (1969) pp. 243–268 |
[8] | J. Simons, "Minimal varieties in Riemannian manifolds" Ann. of Math. , 88 : 1 (1968) pp. 62–105 |
[9] | H. Lawson, "The equivalent Plateau problem and interior regularity" Trans. Amer. Math. Soc. , 173 : 446 (1972) pp. 231–249 |
[10] | 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 |
[11] | A.T. Fomenko, "Bott periodicity from the point of view of an ![]() |
[12] | 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 |
[13a] | 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) |
[13b] | 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) |
[14] | 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 |
[15] | 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 |
[16] | W.Y. Hsiang, H.B. Lawson, "Minimal submanifolds of low cohomogenety" J. Diff. Geom. , 5 : 1 (1971) pp. 1–38 |
[17] | J.E. Brothers, "Invariance of solutions to invariant parametric variational problems" Trans. Amer. Math. Soc. , 262 : 1 (1980) pp. 159–180 |
[18a] | R. Harvey, H.B. Lawson, "On boundaries of complex analytic varieties I" Ann. of Math. , 102 (1975) pp. 233–290 |
[18b] | R. Harvey, H.B. Lawson, "On boundaries of complex analytic varieties II" Ann. of Math. , 106 (1977) pp. 213–238 |
[19] | R. Harvey, H.B. Lawson, "Calibrated geometries" Acta Math. , 148 (1982) pp. 47–157 |
[20] | S.S.-T. Yau, "Kohn–Rossi cohomology and its application to the complex Plateau problem" Ann. of Math. , 113 : 1 (1981) pp. 67–110 |
[21] | 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 |
[22] | A.T. Fomenko, "Plateau problem" , 1–2 , Gordon & Breach (1990) (Translated from Russian) |
Comments
For minimal surfaces in see also [a4], [a5].
References
[a1] | A.T. Fomenko, "Variational principles in topology - Multidimensional minimal surface theory" , Kluwer (1990) (Translated from Russian) |
[a2] | E. Giusti, "Minimal surfaces and functions of bounded variation" , Birkhäuser (1984) |
[a3] | A.T. Fomenko, Dao Chong Tkhi, "Minimal surfaces and Plateau's problem" , Amer. Math. Soc. (Forthcoming) (Translated from Russian) |
[a4] | G. Stolzenberg, "Volumes, limits, and extensions of analytic varieties" , Springer (1966) |
[a5] | E.M. Chirka, "Complex analytic sets" , Kluwer (1989) pp. §19 (Translated from Russian) |
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