Plateau problem

The problem of finding a minimal surface with a given boundary $ \Gamma $. The problem was first formulated by J.L. Lagrange (1760), who reduced it for the class of surfaces of the form $ z= z( x, y) $ to the solution of the Euler–Lagrange equation for a minimal surface. Experiments by J. Plateau (1849) showed that a minimal surface can be obtained in the form of a soap film stretched on a wire framework [1], and the problem has come to be called the Plateau problem.

In a rigorous formulation, the Plateau problem requires some additional refinements relating to the unknown minimal surface and the boundary. For example, one has to determine: whether the solution should be a regular minimal surface or whether it can be sought among generalized minimal surfaces; whether the surface must realise an absolute minimum of the area; what should be the conformal or topological type of the surface; in what sense one should understand the boundary of the surface, etc. The formulation determines the solution and its properties (existence, uniqueness, regularity, etc.), which may differ substantially.

During the 19th century the Plateau problem was solved for certain particular forms of $ \Gamma $, mainly for various polygonal contours (B. Riemann. H.A. Schwarz, K. Weierstrass). In 1928, existence was proved of a solution to the Plateau problem for a generalized minimal surface of disc type, which is represented by Weierstrass' formulas and bounded by a given non-nodal Jordan curve (R. Garnier). In 1931, a solution to the Plateau problem was given in the following formulation (T. Rado): Let $ \Gamma $ be a Jordan curve in $ \mathbf R ^ {n} $, $ n\geq 2 $, then in $ \mathbf R ^ {n} $ there exists a generalized minimal surface, defined in isothermal coordinates $ ( u, v) $ by a position vector $ r= r( u, v) $ that is continuous in the disc $ | w |\leq 1 $, $ w= u+ iv $, and that homeomorphically maps the circle $ | w | = 1 $ onto $ \Gamma $; the area of this generalized minimal surface is the least among all continuous surfaces of disc type stretched on the contour $ \Gamma $, on the assumption that at least one such surface of finite area can be stretched on $ \Gamma $.

After the Plateau problem had been solved in 1931 for a simply-connected surface (J. Douglas), Douglas formulated the so-called Douglas problem on the existence in $ \mathbf R ^ {n} $, $ n \geq 2 $, of a minimal surface having a given topological type (i.e. a given Euler characteristic and orientability character) and being bounded by a given contour $ \Gamma $ consisting of the union of $ k\geq 1 $ Jordan curves $ \Gamma _ {1} \dots \Gamma _ {k} $. In 1936–1940, sufficient solvability conditions for this problem were given, one of which is the possibility of stretching some surface of a given topological type on $ \Gamma $ whose area is less than the area of any surface having a smaller Euler characteristic stretched on the same contour. In that formulation, the Plateau problem was considered and solved also in Riemannian spaces.

In the early 1960-s, a major advance was made in solving the Plateau problem for $ k $-dimensional surfaces, $ k\geq 3 $. Several generalizations of the Plateau problem were proposed, based on new definitions of the concept of a surface, a boundary and area. One of the extensions is based on the following definition of a surface $ X $ and its boundary $ L $ in $ \mathbf R ^ {n} $. Let there be a compact set $ X \subset \mathbf R ^ {n} $, a compact set $ A \subset X $, let $ G $ be an Abelian group, and let $ k \geq 1 $ be an integer; then the Aleksandrov–Čech homology groups $ H _ {k-1} ( A, G) $, $ H _ {k-1} ( X, G) $ are defined, together with the kernel of the homomorphism $ i _ {*} : H _ {k-1} ( A, G) \rightarrow H _ {k-1} ( X, G) $ induced by the imbedding $ i: A\rightarrow X $, which is called the algebraic boundary of $ X $ (in dimension $ k $) relative to $ A $. If $ L $ is a subgroup of $ H _ {k-1} ( A, G) $, then $ X $ is a surface with boundary $ \supset L $ if $ L $ belongs to the algebraic boundary of $ X $; by the area of a compact set $ X $ in $ \mathbf R ^ {n} $ one understands its $ k $-dimensional Hausdorff (spherical) measure $ {\mathcal H} ^ {k} $. Existence and almost-everywhere regularity have been demonstrated for a compact set $ X _ {0} $ that realizes the minimum in the measure $ {\mathcal H} ^ {k} $ over all compacta $ X $ with the given boundary $ L $ subject to these stipulations (as well as topological local Euclidean structure and analyticity). Subsequently, these theorems were extended to the case of surfaces $ X $ in a Riemannian space.

Other proposed generalizations, in particular in terms of integral currents, are in a certain sense equivalent to a formulation in homology terms.

The multi-dimensional Plateau problem (cf. Plateau problem, multi-dimensional) was solved in a classical formulation in 1969 (A.T. Fomenko), when the following theorem was proved: If one is given a $ ( k- 1) $-dimensional submanifold $ \Gamma $, $ k\geq 3 $, in a Riemannian space $ V ^ {n} $, then there exists a surface that realizes a minimum in the Hausdorff measure $ {\mathcal H} ^ {k} $ among all parametrized surfaces $ X $ that are continuous $ f $-transforms in $ V ^ {n} $ of $ k $-dimensional smooth manifolds $ M $ with boundaries homeomorphic to $ \Gamma $ under the mapping $ f: M\rightarrow V ^ {n} $.

Along with the solvability of the Plateau problem, interest attaches to uniqueness and regularity of the solution. Regularity has been most examined. It has been shown that the solution given by Douglas in $ \mathbf R ^ {3} $ does not contain interior branch points. Regularity almost everywhere has been demonstrated for the case of multi-dimensional Plateau problems, and the possibility of the existence of irregular points has been confirmed with examples. As regards uniqueness, only certain sufficient criteria are known (for example, the solution is unique if the given contour $ \Gamma $ has a single-valued convex projection under central or parallel projection onto a certain plane). To emphasise the complexity of this topic, it is sufficient to say that there is reason to expect the existence of smooth Jordan contours spanning a continuum of minimal surfaces of disc type. For a survey of recent results on the Plateau problem see [14].

References

[1]	, Enzyklopaedie der math. Wissenschaften , 2/3 , Teubner (1903)
[2]	G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18
[3]	L. Bianchi, "Vorlesungen über Differentialgeometrie" , Teubner (1910) (Translated from Italian) Zbl 41.0676.01
[4]	R. Courant, "Dirichlet's principle, conformal mapping, and minimal surfaces" , Interscience (1950) (With appendix by M. Schiffer: Some recent developments in the theory of conformal mapping) MR36317
[5]	C. Morrey, "The problem of Plateau on a Riemannian manifold" Ann. of Math. , 49 : 4 (1948) pp. 807–851 MR0027137 Zbl 0033.39601
[6]	T. Radó, "On the problem of Plateau" , Chelsea, reprint (1951) MR0040601 Zbl 0211.13803 Zbl 0007.11804 Zbl 59.1341.01 Zbl 57.0605.11 Zbl 56.0437.01
[7]	J.C.C. Nitsche, "On new results in the theory of minimal surfaces" Bull. Amer. Math. Soc. , 71 (1965) pp. 195–270 MR0173993 Zbl 0135.21701
[8]	R. Osserman, "A proof of the regularity everywhere of the classical solution of Plateau's problem" Ann. of Math. (2) , 91 (1970) pp. 550–569 MR266070
[9]	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
[10]	J.C.C. Nitsche, "The boundary behaviour of minimal surfaces. Kellog's theorem and branch points on the boundary" Invent. Math. , 8 : 4 (1969) pp. 313–333
[11]	R. Osserman, "A survey of minimal surfaces" , v. Nostrand-Reinhold (1969) MR0256278 Zbl 0209.52901
[12]	H. Federer, "Geometric measure theory" , Springer (1969) MR0257325 Zbl 0176.00801
[13]	C. Morrey, "Multiple integrals in the calculus of variations" , Springer (1966) MR0202511 Zbl 0142.38701
[14]	A.T. Fomenko, Dao Chong Tkhi, "Minimal surfaces and Plateau's problem" , Amer. Math. Soc. (Forthcoming) (Translated from Russian)

Comments

In 1973 J.C.C. Nitsche proves the following uniqueness theorem: A regular analytic curve $ \Gamma \subset \mathbf R ^ {3} $ whose total curvature does not exceed the value $ 4 \pi $ bounds precisely one solution surface of Plateau's problem. The bound $ 4 \pi $ is sharp.

References

[a1]	M. Struwe, "Plateau's problem and the calculus of variations" , Princeton Univ. Press (1988) MR992402
[a2]	J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) pp. §455 MR0448224 Zbl 0319.53003