Search

In this paper, we study the asymptotic Plateau problem in [Formula: see text]. We construct the first examples of non-fillable finite curves with no thin tail in [Formula: see text].

We present a novel and comprehensive approach to the study of the parametric Plateau problem for locally strictly convex (LSC) hypersurfaces of prescribed curvature for general convex curvature functions inside general Riemannian manifolds. We prove existence of solutions to the Plateau problem with outer barrier for LSC hypersurfaces of constant or prescribed curvature for general curvature functions inside general Hadamard manifolds modulo a single scalar condition. In particular, convex curvature functions of bounded type are fully treated., Formerly, "The Plateau problem for general curvature functions". Significantly revised for the sake of improved clarity, incorporating notational changes made to arXiv:1002.2982, modifying emphasis on results so as to better reflect the main interests of experts in the field, and tidying up results and removing inelegant technical conditions which have proven themselves to be unnecessary

We prove some existence and non-existence results for complete area minimizing surfaces in the homogeneous space $\mathbb{E}(-1,\tau)$. As one of our main results, we present sufficient conditions for a curve $\Gamma$ in $\partial_{\infty} \mathbb{E}(-1,\tau)$ to admit a solution to the asymptotic Plateau problem, in the sense that there exists a complete area minimizing surface in $\mathbb{E}(-1,\tau)$ having $\Gamma$ as its asymptotic boundary., Comment: Final version, accepted for publication on Ann. Global Anal. Geom. 19 pages, 6 figures

The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank $n \ge 2$ in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite) $n$-cycles of $r^n$ volume growth; prime examples include $n$-cycles associated with $n$-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper CAT(0) spaces of asymptotic rank $n$ extends to a class of $(n-1)$-cycles in the Tits boundaries., Comment: 59 pages. Visual metrics added, minor improvements

Building on work of Du, Gao, and Yau, we give a characterization of smooth solutions, up to normalization, of the complex Plateau problem for strongly pseudoconvex Calabi--Yau CR manifolds of dimension $2n-1 \ge 5$ and in the hypersurface case when $n=2$. The latter case was completely solved by Yau for $n \ge 3$ but only partially solved by Du and Yau for $n=2$. As an application, we determine the existence of a link-theoretic invariant of normal isolated singularities that distinguishes smooth points from singular ones., Comment: 12 pages; v3: to appear in J. Differential Geom

We give a generalization of Meeks-Yau's celebrated embeddedness result for the solutions of the Plateau problem for extreme curves., 15 pages, 6 figures

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space SL ˜ 2 ( R ) with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in SL ˜ 2 ( R ) contained between two bounded entire minimal graphs separated by vertical distance less than 1 + 4 τ 2 π has multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in SL ˜ 2 ( R ) .

We study the minimization problem for the Yang–Mills energy under fixed boundary connection in supercritical dimension n ≥ 5 . We define the natural function space A G in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A G can be also interpreted as a space of weak connections on a “real measure theoretic version” of reflexive sheaves from complex geometry. We prove the existence of weak solutions to the Yang–Mills Plateau problem in the space A G . We then prove the optimal regularity result for solutions of this Plateau problem. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang–Mills density. This generalizes to the general framework of weak L 2 curvatures previous works of Meyer–Riviere and Tao–Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.

We investigate a connection between the two-dimensional Finslerian area functional in arbitrary codimension based on the Busemann–Hausdorff volume form, and well-investigated Cartan functionals to solve the Plateau problem in Finsler spaces. This generalises a previously known result due to Overath and von der Mosel (Manuscripta Math 143(3–4):273–316, 2014) to higher codimension.

This is the second article of the two papers, in which we investigate the holomorphic and formal flattening problem of a non-degenerate CR singular point of a codimension two real submanifold in C n with n ≥ 3 . The problem is motivated from the study of the complex Plateau problem that looks for the Levi-flat hypersurface bounded by a given real submanifold and by the classical complex analysis problem of finding the local hull of holomorphy of a real submanifold in a complex space. The present article is focused on non-degenerate flat CR singular points with at least one non-parabolic Bishop invariant. We will solve the formal flattening problem in this setting. The results in this paper and those in [23] are taken from our earlier arxiv post [22] . We split [22] into two independent articles to avoid it being too long.

We prove a generalization of Reifenberg’s isoperimetric inequality. The main result of this paper is used in Harrison and Pugh (General methods of elliptic minimization, Available on arxiv, 2016) to establish existence of a minimizer for an anisotropically-weighted area functional among a collection of surfaces which satisfies a set of axioms, namely being closed under certain deformations and Hausdorff limits. This problem is known as the axiomatic Plateau problem.

Let $\mathcal{R}$ be an open Riemann surface. In this paper we prove that every continuous function $M \to \mathbb{R}^n$, $n\ge 3$, defined on a divergent Jordan arc $M \subset \mathcal{R}$ can be approximated in the Carleman sense by conformal minimal immersions; thus providing a new generalization of Carleman's theorem. In fact, we prove that this result remains true for null curves and many other classes of directed holomorphic immersions for which the directing variety satisfies a certain flexibility property. Furthermore, the constructed immersions may be chosen to be complete or proper under natural assumptions on the variety and the continuous map. As a consequence we give an approximate solution to a Plateau problem for divergent Jordan curves in the Euclidean spaces., 31 pages, 2 figures

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by Lytchak--Wenger, which satisfies a related maximality condition., Comment: 33 pages

Using the classical approach we show the existence of disc type solutions to the asymptotic Plateau problem in certain Hadamard manifolds which may have arbitrarily strong curvature and volume growth.

We show that for any Jordan curve Gamma in S-infinity(2) (H-3) with at least one smooth point, there exists an embedded H-plane P-H in H-3 with partial derivative P-infinity(H) = Gamma for any H is an element of [0, 1)., Fulbright Grant; Scientific and Technological Research Council of Turkey (TÜBİTAK)) 2219 Grant

We study the asymptotic Dirichlet and Plateau problems on Cartan-Hadamard manifolds satisfying the so-called Strict Convexity (abbr. SC) condition. The main part of the paper consists in studying the SC condition on a manifold whose sectional curvatures are bounded from above and below by certain functions depending on the distance to a fixed point. In particular, we are able to verify the SC condition on manifolds whose curvature lower bound can go to -infinity and upper bound to 0 simultaneously at certain rates, or on some manifolds whose sectional curvatures go to -infinity faster than any prescribed rate. These improve previous results of Anderson, Borb\'ely, and Ripoll and Telichevsky. We then solve the asymptotic Plateau problem for locally rectifiable currents with Z_2-multiplicity in a Cartan-Hadamard manifold satisfying the SC condition given any compact topologically embedded (k-1)-dimensional submanifold of \partial_{\infty}M, 2\leq k\leq n-1, as the boundary data. We also solve the asymptotic Plateau problem for locally rectifiable currents with Z-multiplicity on any rotationally symmetric manifold satisfying the SC condition given a smoothly embedded submanifold as the boundary data. These generalize previous results of Anderson, Bangert, and Lang. Moreover, we obtain new results on the asymptotic Dirichlet problem for a large class of PDEs. In particular, we are able to prove the solvability of this problem on manifolds with super-exponential decay (to -infinity) of the curvature., Comment: This is a corrected and improved version. We thank Urs Lang for his valuable comments on previous versions, in particular, for pointing out an incorrect assumption in Theorem 1.5 in the first version

We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used to control the shapes of all curves and therefore the geometry and topology of the original metric space. The class of spaces arising in this way as intrinsic minimal discs is a natural generalization of the class of Ahlfors regular discs, well-studied in analysis on metric spaces., typos corrected, references added, minor improvements to the exposition

Groping our way toward a theory of singular spaces with positive scalar curvatures we look at the Dirac operator and a generalized Plateau problem in Riemannian manifolds with corners. Using these, we prove that the set of C 2-smooth Riemannian metrics g on a smooth manifold X, such that scalg(x) ≥ κ(x), is closed under C 0-limits of Riemannian metrics for all continuous functions κ on X. Apart from that our progress is limited but we formulate many conjectures. All along, we emphasize geometry, rather than topology of manifolds with their scalar curvatures bounded from below.

We provide new general methods in the calculus of variations for the anisotropic Plateau problem in arbitrary dimension and codimension. A new direct proof of Almgren's 1968 existence result is presented; namely, we produce from a class of competing "surfaces," which span a given bounding set in some ambient space, one with minimal anisotropically weighted area. In particular, rectifiability of a candidate minimizer is proved without the assumption of quasiminimality. Our ambient spaces are a class of Lipschitz neighborhood retracts which includes manifolds with boundary and manifolds with certain singularities. Our competing surfaces are rectifiable sets which satisfy any combination of general homological, cohomological or homotopical spanning conditions. An axiomatic spanning criterion is also provided. Our boundaries are permitted to be arbitrary closed subsets of the ambient space, providing a good setting for surfaces with sliding boundaries., Axiomatic spanning conditions clarified, recent history added with comparisons, details added to proofs, notation simplified, generalization of main result to elliptic measures deferred to a separate paper

We prove a compactness principle for the anisotropic formulation of the Plateau problem in any codimension, in the same spirit of the previous works of the authors \cite{DelGhiMag,DePDeRGhi,DeLDeRGhi16}. In particular, we perform a new strategy for the proof of the rectifiability of the minimal set, based on the new anisotropic counterpart of the Allard rectifiability theorem proved by the authors in \cite{DePDeRGhi2}. As a consequence we provide a new proof of Reifenberg existence theorem.

We prove that if an asymptotically Schwarzschildean 3-manifold (M,g) contains a properly embedded stable minimal surface, then it is isometric to the Euclidean space. This implies, for instance, that in presence of a positive ADM mass any sequence of solutions to the Plateau problem with diverging boundaries can never have uniform height bounds, even at a single point. An analogous result holds true up to ambient dimension seven provided polynomial volume growth on the hypersurface is assumed., Comment: Final version, to appear on Calc. Var. Partial Differential Equations

We prove a compactness principle for the anisotropic formulation of the Plateau problem in codimension one, along the same lines of previous works of the authors [DGM14, DPDRG15]. In particular, we perform a new strategy for proving the rectifiability of the minimal set, avoiding the Preiss' Rectifiability Theorem [Pre87]., Comment: 14 pages

Given a domain $\Omega$ of $\mathbb{R}^{m+1}$ and a $k$-dimensional non-degenerate minimal submanifold $K$ of $\pa \Omega$ with $1\le k\le m-1$, we prove the existence of a family of embedded constant mean curvature hypersurfaces which as their mean curvature tends to infinity concentrate along $K$ and intersecting $\partial \Omega$ perpendicularly., Comment: 28 pages

We prove a new existence theorem pertaining to the Plateau problem in 3-dimensional Euclidean space. We compare the approach of E.R. Reifenberg with that of H. Federer and WH. Fleming. A relevant technical step consists in showing that compact rectifiable surfaces are approximatable in Hausdorff measure and in Hausdorff distance by locally acyclic surfaces having the same boundary

Using a standard fact in hyperbolic geometry, we give a simple proof of the uniqueness of PL minimal surfaces, thus filling in a gap in the original proof of Jaco and Rubinstein. Moreover, in order to clarify some ambiguity, we sharpen the definition of PL minimal surfaces, and prove a technical lemma on the Plateau problem in the hyperbolic space.

This paper proposes a direct approach to solve the Plateau's problem in codimension higher than one. The problem is formulated as the minimization of the Hausdorff measure among a family of d-rectifiable closed subsets of R n : following the previous work [13] , the existence result is obtained by a compactness principle valid under fairly general assumptions on the class of competitors. Such class is then specified to give meaning to boundary conditions. We also show that the obtained minimizers are regular up to a set of dimension less than ( d − 1 ) .

The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of area-minimizing mappings of the disk. Specifically, we find, for $K \in \mathbb N,$ $k_1,...,k_K\in \mathbb N$ with sum $Q$ and any collection of $K$ disjoint Lipschitz Neighborhood Retract Jordan curves, optimal multiple-valued boundary data with these multiplicities which extends to a Dirichlet minimizing $Q$-valued function with minimal Dirichlet energy among all possible monotone parameterizations of the boundary curves. Under a condition analogous to the Douglas condition for minimizers from planar domains, conformality of the minimizer follows from topological methods and some complex analysis. Finally, we analyze two particular cases: in contrast to single-valued Douglas solutions, we first give a class of examples for which our multiple-valued Plateau solution has branch points. Second, we give examples of a degenerate behavior, illustrating the weakness of the multiple-valued maximum principle and provide motivation for our analogous Douglas condition., Comment: 18 pages, 2 figures, changed the proof of Theorem 1.2

We prove existence and regularity of minimizers for H\"older densities over general surfaces of arbitrary dimension and codimension in \(\R^n \), satisfying a cohomological boundary condition, providing a natural dual to Reifenberg's Plateau problem. We generalize and extend methods of Reifenberg, Besicovitch, and Adams, in particular we generalize a particular type of minimizing sequence used by Reifenberg (whose limits have nice properties, including lower bounds on lower density and finite Hausdorff measure,) prove such minimizing sequences exist, and develop cohomological spanning conditions. Our cohomology lemmas are dual versions of the homology lemmas in the celebrated appendix by Adams found in Reifenberg's 1960 paper., Comment: 39 pages, 2 figures, to appear. Lipschitz conditions are replaced by Holder ones. A stronger regularity result and a discussion of the Adams surface are included, Calculus of Variations and Partial Differential Equations, 2016

In this article, we prove the existence of two embedded minimal annuli in a slab which are all bounded by a Jordan convex curve and a straight line.

for some constant K, where κ[M ] = (κ1, . . . , κn) denotes the principal curvatures of M . Important examples include the classical Plateau problem for minimal or constant mean curvature surfaces and the corresponding problem for Gauss curvature, which was treated recently by the authors [12] and independently by Trudinger-Wang [24]. In this paper as in our previous work [12], we are concerned with locally convex hypersurfaces. Accordingly, the function f is assumed to be defined in the convex cone Γn ≡ { λ ∈ R : each component λi > 0 } in R and satisfy the fundamental structure conditions

We study Plateau's problem for two-dimensional parametric integrals F(X) := ∫ B F(X, X u Λ X v ) du dv, the Lagrangian F(x, z) of which is positive definite and at least semi-elliptic. It turns out that there always exists a conformally parametrized minimizer. Any such minimizer X is seen to be Holder-continuous in the parameter domain B and continuous up to its boundary. If F possesses a perfect dominance function G of class C 2 , we can establish higher regularity of X in the interior. In fact, we prove X ∈ H 2,2 loc (B, R n ) ∩ C 1,σ (B, R n ) for some a > 0. Finally, we discuss the existence of perfect dominance functions.

We solve the asymptotic Plateau problem in every Gromov hyperbolic Hadamard manifold (X,g) with bounded geometry. That is, we prove existence of complete (possibly singular) k-dimensional area minimizing surfaces in X with prescribed boundary data at infinity, for a large class of admissible limit sets and for all $2 \le k < dim X$ . The result also holds with respect to any riemannian metric $\tilde g$ on X which is lipschitz equivalent to g.

We consider the complex Plateau problem for strongly pseudoconvex contours in non-Kahler manifolds. We give a necessary and sufficient condition for the existence of solution in the class of manifolds carrying pluriclosed metric forms and propose a conjecture for the general case. 0. Introduction. Recall that the complex Plateau problem for a compact real submanifold M of a complex manifold X consists in finding an analytic chain A ⊂ X \ M with “boundary” M . More precisely, one wants to find a complex analytic subset A ⊂ X \ M such that ∂[A] = [M ] in the sense of currents. A necessary condition on M for the complex Plateau problem to have a solution is the maximal complexity of M , i.e. M should be a CR-submanifold of X with dimCR M = p, where dimR M = 2p + 1. In the case when X is Stein this is also sufficient (see [H]). Already in the case X = CP the maximal complexity of the “contour” M is not sufficient any more [Db]. We refer the interested reader to [Db-H] for an extensive exposition on the Plateau problem in projective space. In this paper we restrict ourselves to strongly pseudoconvex contours M , which are already the boundaries of some abstract complex manifolds. At the same time we look for solutions of the complex Plateau problem in more general ambient manifolds. Let w be a strictly positive, smooth (1, 1)-form on the complex manifold X. Definition 0.1. We say that w is pluriclosed if ddw = 0. The Hermitian metric canonically associated with such a w is often also called pluriclosed . Denote by ∆ the unit disk in C. Definition 0.2. We say that a complex space X is disk-convex in dimension k if for any compact set K ⋐ X there is another compact set K 1991 Mathematics Subject Classification: Primary 32D15.

We do a historical review of the surfaces with constant mean curvature motivating its study by the classical isoperimetric problem. We describe the two ways that historically have gone parallel, depending if in the variational characterization we consider only one constraint (the volume) or two constrains (the volume and the boundary). In the first case, we recall the different characterizations of the sphere in the family of closed surfaces with constant mean curvature. The absence of new examples brought as a consequence a great effort in the seek of new surfaces and techniques that succeeded in the discovery by Wente of a torus immersed in Euclidean space with constant mean curvature. This opened news sights of the theory. For surfaces with boundary, Douglas and Rado considered the Plateau problem for minimal surfaces and later, in the general case of mean curvature, by Heinz, Hildebrandt and others in the fifties of XX century.

In this paper, we give some examples of area-minimizing surfaces to clarify some wellknown features of these surfaces in more general settings. The first example is about Meeks–Yau’s result on the embeddedness of the solution to the Plateau problem. We construct an example of a simple closed curve in R3 which lies in the boundary of a mean convex domain in R3, but the area-minimizing disk in R3 bounding this curve is not embedded. Our second example shows that White’s boundary decomposition theorem does not extend when the ambient space has nontrivial homology. Our last examples show that there are properly embedded absolutely area-minimizing surfaces in a mean convex 3-manifold M such that, while their boundaries are disjoint, they intersect each other nontrivially, unlike the area-minimizing disks case., EU-FP7; Scientific and Technological Research Council of Turkey (TÜBİTAK)

In this chapter we shall discuss minimal graphs, i.e. graphs whose area has vanishing first variation. After introducing the variational equations, we shall work on the existence of minimal graphs with prescribed boundary (Plateau problem), their uniqueness, stability and regularity in codimension 1.

We show that for a very general class of curvature functions defined in the positive cone, the problem of finding a complete strictly locally convex hypersurface in $H^n+1$ satisfying $f(\kappa)=\sigma\in(0, 1)$ with a prescribed asymptotic boundary $\Gamma$ at infinity has at least one smooth solution with uniformly bounded hyperbolic principal curvatures. Moreover if $\Gamma$ is (Euclidean) starshaped, the solution is unique and also (Euclidean) starshaped while if $\Gamma$ is mean convex the solution is unique. We also show via a strong duality theorem that analogous results hold in De Sitter space. A novel feature of our approach is a "global interior curvature estimate"., Comment: 20 pages

We provide a new proof of the classical result that any closed rectifiable Jordan curve Gamma in space being piecewise of class C^2 bounds at least one immersed minimal surface of disc-type, under the additional assumption that the total curvature of Gamma is smaller than 6*Pi. In contrast to the methods due to Osserman, Gulliver and Alt, our proof relies on a polygonal approximation technique, using the existence of immersed solutions of Plateau's problem for polygonal boundary curves, provided by the first author's accomplishment of Garnier's ideas., Comment: 12 pages

We study the Plateau Problem of finding an area minimizing disk bounding a given Jordan curve in Alexandrov spaces with curvature $≥ κ$. These are complete metric spaces with a lower curvature bound given in terms of triangle comparison. Imposing an additional condition that is satisfied by all Alexandrov spaces according to a conjecture of Perel’man, we develop a harmonic map theory from two dimensional domains into these spaces. In particular, we show that the solution to the Dirichlet problem from a disk is Holder continuous in the interior and continuous up to the boundary. Using this theory, we solve the Plateau Problem in this setting generalizing classical results in Euclidean space (due to J. Douglas and T. Rado) and in Riemannian manifolds (due to C.B. Morrey).

In this chapter we shall present another solution of the general problem of Plateau, i.e. of the Douglas problem. This is the question whether a configuration 〈Γ 1,Γ 2,…,Γ k 〉 of several closed curves Γ j may bound multiply connected minimal surfaces that could be of higher genus and even nonorientable; for a special case this problem was studied in Vol. 1. As background, an approach to Teichmuller theory of compact oriented Riemann surfaces is outlined which then is extended to oriented Riemann surfaces with boundary. These results are basic for the variational treatment of the general Plateau problem, as one has to deal with variations of the complex structure of two-manifolds. The second basic tool is a compactness theorem of Mumford which is combined with Courant’s condition of cohesion to obtain nondegenerate solutions of the variational problem connected with the general Plateau problem. Then the crucial ideas for solving this problem are outlined and some of the principal existence results due to Douglas, Courant, and Shiffman are stated. Finally, the general Plateau problem for orientable surfaces is handled in combination with an obstacle problem in order to obtain a sufficient condition for the solvability of the general Douglas–Plateau problem that can be formulated in geometric-topological terms.

We describe a novel technique for solving the Plateau problem for constant curvature hypersurfaces based on recent work of Harvey and Lawson. This is illustrated by an existence theorem for hypersurfaces of constant Gaussian curvature in \({\mathbb{R}^{n+1}}\).

Douglas and Rado showed that an embedded rectifiable curve bounds a disk of least area. It is shown that this result holds for all rectifiable curves, including singular ones.

In this paper we prove theorems for convergence of discrete solutions of the Plateau problem under the assumption that the contour is rectifiable.

Let X be a compact connected strongly pseudoconvex CR Manifold of real dimension 2n − 1 in C n+1 . Tanaka introduced a spectral sequence E (p,q) r (X) with E (p,q) 1 (X) being the Kohn-Rossi cohomology group and E (k,0) 2 (X) being the holomorphic De Rham cohomology denoted by H k h (X). We study the holomorphic De Rham cohomology in terms of the s-invariant of the isolated singularities of the variety V bounded by X. We give a characterization of the singularities with vanishing s-invariants. For n ≥ 3, Yau used the Kohn-Rossi cohomology groups to solve the classical complex Plateau problem in 1981. For n = 2, the problem has remained unsolved for over a quarter of a century. In this paper, we make progress in this direction by putting some conditions on X so that V will have very mild singularities. Specifically, we prove that if dim X = 3 and H 2 h(X) = 0, then X is a boundary of complex variety V with only isolated quasi-homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational.

The aim of these pages is to give a brief, self-contained introduction to that part of Geometric Measure Theory which is more directly related to the Calculus of Variations, namely the theory of currents and its applications to the solution of Plateau problem. (The theory of finite perimeter sets, which is closely related to currents and to the Plateau problem, is treated in the article “Free interfaces and free discontinuities: variational problems”). Named after the belgian physicist J.A.F. Plateau (1801-1883), this problem was originally formulated as follows: find the surface of minimal area spanning a given curve in the space. Nowadays, it is mostly intended in the sense of developing a mathematical framework where the existence of k-dimensional surfaces of minimal volume that span a prescribed boundary can be rigorously proved. Indeed, several solutions have been proposed in the last century, none of which is completely satisfactory. One difficulty is that the infimum of the area among all smooth surfaces with a certain boundary may not be attained. More precisely, it may happen that all minimizing sequences (that is, sequences of smooth surfaces whose area approaches the infimum) converge to a singular surface. Therefore one is forced to consider a larger class of admissible surfaces than just smooth ones (in fact, one might want to do this also for modelling reasons—this is indeed the case with soap films, soap bubbles, and other capillarity problems). But what does it mean that a set “spans” a given curve? and what should we intend by area of a set which is not a smooth surface? The theory of integral currents developed by H. Federer and W.H. Fleming [4] provides a class of generalized (oriented) surfaces with well-defined notions of boundary and area (called mass) where the existence of minimizers can

Let $(M,Q)$ be a compact, three dimensional manifold of strictly negative sectional curvature. Let $(\Sigma,P)$ be a compact, orientable surface of hyperbolic type (i.e. of genus at least two). Let $\theta:\pi_1(\Sigma,P)\to\pi_1(M,Q)$ be a homomorphism. Generalising a recent result of Gallo, Kapovich and Marden concerning necessary and sufficient conditions for the existence of complex projective structures with specified holonomy to manifolds of non-constant negative curvature, we obtain necessary conditions on $\theta$ for the existence of a so called $\theta$-equivariant Plateau problem over $\Sigma$, which is equivalent to the existence of a strictly convex immersion $i:\Sigma\to M$ which realises $\theta$ (i.e. such that $\theta=i_*$)., Comment: 47 pages, 2 figures, considerably shortened version containing much the same content as before

In the first part of this paper we prove that certain functionals of Ginzburg-Landau type for maps from a domain in R n+k into R k converge in a suitable sense to the area func- tional for surfaces of dimension n (Theorem 1.1). In the second part we modify this result in order to include Dirichlet boundary condition (Theorem 5.5), and, as a corollary, we show that the rescaled energy densities and the Jacobians of minimizers con- verge to minimal surfaces of dimension n (Corollaries 1.2 and 5.6). Some of these results were announced in (2).