Search

The most famous problem in the theory of minimal surfaces is the so called Plateau problem, where one is looking for a minimal surface spanning a given boundary. This is a problem with a fixed boundary and was essentially solved around 1930 by Douglas and Rado.

Publisher Summary The chapter discusses the equivariant loop theorem for three-dimensional manifolds that is needed in settling the Smith conjecture and reviews the existence theorems for minimal surfaces. The equivariant version of the loop theorem says that there are a finite number of properly embedded disks in M that satisfy the required properties. The loop theorem respects the action of the group G in a suitable manner. The chapter puts a metric on M so that the group G acts isometrically and so that ∂M is convex with respect to the outward normal. Then with respect to this metric, the existence of an immersed disk D 1 is demonstrated, in M whose boundary ∂ D 1 , represents a nontrivial element in π 1 ( S ) and whose area is minimal among all such disks. The chapter describes Morrey's solution for the plateau problem in a general Riemannian manifold. The existence theorem for manifolds with boundary proves that M is homogeneously regular if M is a subdomain of another homogeneously regular manifold N that has no boundary.

This paper presents a numerical method for finding the solution of Plateau's problem in parametric form. Using the properties of minimal surfaces we succeded in transferring the problem of finding the minimal surface to a problem of minimizing a functional over a class of scalar functions. A numerical method of minimizing a functional using the first variation is presented and convergence is proven. A numerical example is given.

We show that in the setting of proper metric spaces one obtains a solution of the classical two-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area (in the sense of convex geometry) has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hoelder exponents of some area-minimizing discs., typos corrected, references updated

We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y, dY ). We say that a metric space (Y, dY ) is a quasiconformal Jordan domain if the completion ̄Y of (Y, dY ) has finite Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism ϕ: 𝔻 →(Y, dY ) that is quasiconformal in the geometric sense. We show that ϕ has a continuous, monotone, and surjective extension Φ: 𝔻 ̄ → Y ̄. This result is best possible in this generality. In addition, we find a necessary and sufficient condition for Φ to be a quasiconformal homeomorphism. We provide sufficient conditions for the restriction of Φ to 𝕊1 being a quasisymmetry and to ∂Y being bi-Lipschitz equivalent to a quasicircle in the plane.

We present the novel method for generation of periodic surfaces based on the simple Landau-Ginzburg model of microemulsion. We test the method on four minimal surfaces (P,D,G, and I-WP), find two new surfaces of cubic symmetry, show how to obtain periodic surfaces of high genus and n-tuply-continuous phases. We point that the Landau model used here should be generic for all systems characterized by internal interfaces, including the diblock copolymer systems., Comment: 15 pages, Plain TeX , figures available on request or at http://saka.ichf.edu.pl/WTGRH/surfaces.html

Let $(M,g, k)$ be an initial data set for the Einstein equations of general relativity. ¶ We prove that there exist solutions of the Plateau problem for marginally outer trapped surfaces (MOTSs) that are stable in the sense of MOTSs. This answers a question of G. Galloway and N. O’Murchadha raised in “Some remarks on the size of bodies and black holes,” [Classical Quantum Gravity 25 (2008), no. 10, 105009, 9. MR 2416045] and is an ingredient in the proof of the spacetime positive mass theorem given by L.-H. Huang, D. Lee, R. Schoen, and the first named author. ¶ We show that a canonical solution of the Jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to a given end, provided that this complement contains no weakly past outer trapped regions. The graph of this solution relates the area of the horizon to the global geometry of the initial data set in a non-trivial way. We prove the existence of a Scherk-type solution of the Jang equation outside the union of all weakly future or past outer trapped regions in the initial data set. This result is a natural exterior analogue for the Jang equation of the classical Jenkins–Serrin theory. ¶ We extend and complement existence theorems for Scherk–type constant mean curvature graphs over polygonal domains in $(M,g)$, where $(M,g)$ is a complete Riemannian surface. We can dispense with the a priori assumptions that a sub solution exists and that $(M,g)$ has particular symmetries. Also, our method generalizes to higher dimensions.

Using ideas suggested by some recent developments in string theory, we give here an elementary demonstration of one of the key steps in Douglas' celebrated proof of the existence of solutions of the Plateau problem in n dimensions.

It was observed by P. Lelong [16] that a pure k-dimensional analytic subvariety V of an open set Q contained in Cm determines a current [ V] on Q via integration over the regular points of V. The current [ V] has several special properties: It is of type (m k, m k), positive, d-closed, and locally rectifiable. Conversely, J. King [14] proved that every current with these properties corresponds to integration over a pure k-dimensional variety with positive integral multiplicities. In this paper, we generalize King's theorem by dropping the assumption that the current be positive and then concluding (with an added assumption on the size of the support) that the current corresponds to integration over a pure k-dimensional variety with arbitrary integral multiplicities (Theorem 2.1). Such currents, which represent analytic cycles of dimension k, are called holomorphic k-chains (Definition 1.7) and are characterized by Theorem 2.1. Our techniques involve Federer's geometric theory of currents [7]. In Section 1.1 we describe the locally rectifiable currents of Federer and Fleming [8], for which we provide a new characterization (Theorems 1.4 and 1.5). Using Theorems 1.5 and 2.1, we can also characterize the holomorphic k-chains as the d-closed currents of type (m k, m k) that are representable by integration and have positive integral 2k-densities almost everywhere (Theorem 2.2). Our characterization Theorem 2.1 can be used to prove that all stable, d-closed, locally rectifiable currents in CPm (complex projective m-space) are algebraic cycles. This application, which inspired this paper, is due to Lawson and Simons [15]. We also prove that if a complex subvariety is a solution to a Plateau problem in Cm, then it is the only solution (Theorem 3.6). As a special case of Theorem 2.1 we obtain a new, somewhat simpler proof of King's theorem characterizing the positive holomorphic k-chains. (Another proof and generalization of King's theorem is given in [11], which uses a result of E. Bombieri [3], [4].) We also use our characterization theorem to deduce a theorem of E. Bishop [1] on extending analytic subvarieties with finite volume.

The chemical potential at a curved or faceted surface is proportional to the surface divergence of the previousll-defined vector function ξ, lhich represents the energetics of both isotropic and anisotropic surfaces in a convenient manner. This relationship is sholn to be a generaliled reformulation of the eluations of Gibbs, Thompson, Herring and Johnson, but leads in addition to nel alternative formulas in lhich certain underlling slmmetries become evident. Applications of the vector formalism include analltic proof of the lulff theorem, distribution of torlue on a facet, formulation of the anisotropic Plateau problem, and the eluilibrium shapes of particles at surfaces and grain boundaries.

The problem of Plateau or the proof of the existence of a surface of least area bounded by a given Jordan contour in Euclidean space is one of the classical problems of the Calculus of Variations. This problem has been studied at least since the time of Riemann but it was not until 1930 that satisfactory proofs for a general contour were given independently by Douglas [25]' and Rado [15]. However, before that time, considerable information was gathered and the problem was solved in many interesting special cases. For the literature of this period, we refer to the excellent report of Rad6 [38] on the problem of Plateau. Shortly after the appearance of the solutions of Douglas and Rado, McShane presented an interesting solution which was perhaps more along the lines of the straightforward "direct methods of the Calculus of Variations" (see [30]). In the past seventeen years the problem has been generalized by replacing r by several contours r1, * * *, rk and by prescribing the topological structure of the desired surface ([5], [19], [27], [28], 129], [33], 135], [41], [43]), or by replacing the condition that the surface be bounded by a given contour by the condition that only parts of the boundary be prescribed, the remainder of the boundary being merely restricted to lie along certain manifolds (see [18], [22], [23], [24]). Finally minimal surfaces which do not furnish even relative minima have been discussed (see [20], [21], [31], [32], [33], [34], [35], [39], [40], [42], [43]). The present paper generalizes the problem by replacing the underlying Euclidean space by a Riemannian manifold of considerable generality. The writer knows of only one paper in which the Plateau problem has been solved in any space other than Euclidean space, namely the recent paper by A. Lonseth [7] where the space considered is "hyperbolic space". However, S. Bochner [2] has studied the "harmonic" functions in a general Riemannian metric (of sufficient differentiability) but has not demonstrated their minimizing property in general. This work is, of course of great interest as it is still true in a Riemannian space (of sufficient differentiability) that if a minimal surface is mapped conformally on a plane region, then the representing vector is "harmonic" in Bochner's sense and also that a surface of least area (if of sufficient differentiability) is a minimal surface (in the sense of Differential Geometry). The present writer does not attempt to generalize all of the preceding results on the Plateau problem but restricts himself to the case of a surface of the type of a k-fold connected plane region bounded by k given contours. The space is a "homogeneously regular" (see Def. 3.1) Riemannian manifold 9) of class C'. Every compact, regular Riemannian manifold of class C' is seen to be homo

Since the earlier works on minimal surfaces, both the geometrical and the analytical aspects of the theory and their interrelations have been of profound interest for the theory itself and for its connections with numerous branches of analysis and differential geometry. When the surface was considered in the small, it was assumed to be regular at every point-;by regularity, we mean that the first fundamental form of the surface is positive definite. Bonnet proved the theorem that the Gauss mapping of the surface on the unit sphere in R3 is conformal. The analytical representation of minimal surfaces by the Weierstass formula was of fundamental importance for the investigation of the Plateau problem with polygonal boundaries. In his famous paper [1]1 H. A. Schwarz established analytically a surface with a quadrilateral as boundary and extended it into the space RK. For us it is significant that Schwarz formulated the problem to find a minimal surface, which is simply connected and without singular points in the interior. In view of the differential geometry the last requirement is important and natural. In his solution the Gauss mapping is one to one except at the vertices. This fact, it seems to me, was not of minor importance for the success of his method. Modern achievements in the Plateau problem [5, 6, 7, 8, 9] propose a great variety of analytical and geometrical problems of importance. As we are not going to be concerned with the Plateau problem in general, nor with the modern analytical methods, we shall mention here only certain geometrical aspects of the theory in connection with singular points. Until the work of A. Haar [2] the surfaces considered were in R3 (x, y, z) and capable of representation by z = z(x, y). Accordingly the boundary curve was assumed to have a convex shaped projection. Rad6 [4] proved the theorem that any minimal surface whose boundary curve possesses a convex or star shaped projection, has no branch points in the interior. In the formulation and treatment of the Plateau-Douglas problem initiated by J. Douglas [5] and independently, but less general, by T. Rad6 [7], the condition that the surface be regular in the interior is dropped. Since the derivatives are continuous in the interior, the only possible singular points are branch points where all the derivatives are zero. In view of the generality of the contour admitted and the topological structure of the surface considered, regularity in general is not to be expected; or at least one has to count on possible self-intersections of the surface. Rado [3] emphasized the distinction