Search

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

We study geometric properties of compact stable minimal surfaces with boundary in homogeneous 3-manifolds $X$ that can be expressed as a semidirect product of $\mathbb{R}^2$ with $\mathbb{R}$ endowed with a left invariant metric. For any such compact minimal surface $M$, we provide a priori radius estimate which depends only on the maximum distance of points of the boundary $\partial M$ to a vertical geodesic of $X$. We also give a generalization of the classical Rado's Theorem in $\mathbb{R}^3$ to the context of compact minimal surfaces with graphical boundary over a convex horizontal domain in $X$, and we study the geometry, existence and uniqueness of this type of Plateau problem., 31 pages, 3 figures. Comments are welcome

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

In this article we treat two closely related problems: 1) the upper semi-continuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the $2$-dimensional minimal cones in $\mathbb R^3$. ¶ Given an open set $\Omega\subset\mathbb R^n$, a closed set $E\subset \Omega$ is said to be Almgren minimal of dimension $d$ in $\Omega$ if it minimizes the $d$-Hausdorff measure among all its Lipschitz deformations in $\Omega$. We say that a $d$-dimensional minimal set $E$ in an open set $\Omega$ admits upper semi-continuity if, whenever $\{f_n(E)\}_n$ is a sequence of deformations of $E$ in $\Omega$ that converges to a set $F$, then we have ${\mathcal H}^d(F)\ge \limsup_n {\mathcal H}^d(f_n(E))$. This guarantees in particular that $E$ minimizes the $d$-Hausdorff measure, not only among all its deformations, but also among limits of its deformations. ¶ As proved in [19], when several $2$-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known $2$-dimensional minimal cones in $\mathbb R^n$ to generate new families of minimal cones by taking their almost orthogonal unions. ¶ The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].

We study the Plateau problem with a lower-dimensional obstacle in R-n. Intuitively, in R-3 this corresponds to a soap film (spanning a given contour) that is pushed from below by a "vertical" 2D half-space (or some smooth deformation of it). We establish almost optimal C-1,C- 1/2- estimates for the solutions near points on the free boundary of the contact set, in any dimension n >= 2. The C-1,C-1/2- estimates follow from an epsilon-regularity result for minimal surfaces with thin obstacles in the spirit of the De Giorgi's improvement of flatness. To prove it, we follow Savin's small perturbations method. A nontrivial difficulty in using Savin's approach for minimal surfaces with thin obstacles is that near a typical contact point the solution consists of two smooth surfaces that intersect transversally, and hence it is not very flat at small scales. Via a new "dichotomy approach" based on barrier arguments we are able to overcome this difficulty and prove the desired result., Journal für die reine und angewandte Mathematik, 767, ISSN:0075-4102, ISSN:1435-5345

We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in $${\mathbb H}^2\times {\mathbb R}$$ . In particular, we identify the collection of Jordan curves in $$\partial _\infty ({\mathbb H}^2\times {\mathbb R})$$ which bounds an area minimizing surface in $${\mathbb H}^2\times {\mathbb R}$$ . Furthermore, we study the similar problem for minimal surfaces, and show that the situation is highly different.

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.

Let $\Omega \subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial \Omega$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial \Omega \to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial \Omega$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\bar{\Omega}$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial \Omega$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\bar{\Omega}$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial \Omega$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.

We prove the existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature in Cartan-Hadamard manifolds $N$. More precisely, given a suitable subset $L$ of the asymptotic boundary of $N$ and a suitable function $H$ on $N$, we are able to construct a set of locally finite perimeter whose boundary has generalized mean curvature $H$ provided that $N$ satisfies the so-called strict convexity condition and that its sectional curvatures are bounded from above by a negative constant. We also obtain a multiplicity result in low dimensions., Comment: The proof of Lemma 2.1 is rewritten providing a quantitative estimate for the decay of the functional ${\mathcal F}$. To appear in Proceedings of the AMS

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

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.

Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. It has been an interesting question to find an intrinsic smoothness criteria for the complex Plateau problem. For $n\ge 3$ and $N=n+1$, Yau found a necessary and sufficient condition for the interior regularity of the Harvey-Lawson solution to the complex Plateau problem by means of Kohn--Rossi cohomology groups on $X$ in 1981. For $n=2$ and $N\ge n+1$, the first and third authors introduced a new CR invariant $g^{(1,1)}(X)$ of $X$. The vanishing of this invariant will give the interior regularity of the Harvey-Lawson solution up to normalization. For $n\ge 3$ and $N>n+1$, the problem still remains open. In this paper, we generalize the invariant $g^{(1,1)}(X)$ to higher dimension as $g^{(\Lambda^n 1)}(X)$ and show that if $g^{(\Lambda^n 1)}(X)=0$, then the interior has at most finite number of rational singularities. In particular, if $X$ is Calabi--Yau of real dimension $5$, then the vanishing of this invariant is equivalent to give the interior regularity up to normalization., Comment: arXiv admin note: substantial text overlap with arXiv:1203.1380

This is the first article of the two papers in which we investigate the holomorphic and formal flattening problem for a codimension two real submanifold in $${\mathbb C}^n$$ with $$n\ge 3$$ near a non-degenerate CR singular point. The problem is motivated from the study of the complex Plateau problem that seeks for the Levi-flat hypersurface bounded by a given real submanifold and is motivated 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 the case of CR singular points with at least one elliptic direction. We solve the holomorphic flattening problem and thus provide a complete description of the local hull of holomorphy in this setting. The results in this paper and those in (Flattening of CR singular points and analyticity of the local hull of holomorphy II, p. 60, 2014) are taken from our arxiv post (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012). We split (Flattening of CR singular points and analyticity of the local hull of holomorphy, 2012) into two independent articles to avoid it being too long.

The singly periodic Scherk surfaces with higher dihedral symmetry have [math] -ends that come together based upon the value of [math] . These surfaces are embedded provided that [math] . Previously, this inequality has been proved by turning the problem into a Plateau problem and solving, and by using the Jenkins–Serrin solution and Krust’s theorem. In this paper we provide a proof of the embeddedness of these surfaces by using some results about univalent planar harmonic mappings from geometric function theory. This approach is more direct and explicit, and it may provide an alternate way to prove embeddedness for some complicated minimal surfaces.

In mathematics, the classical Plateau problem consists of finding the surface of least area that spans a given rigid boundary curve. A physical realization of the problem is obtained by dipping a stiff wire frame of some given shape in soapy water and then removing it; the shape of the spanning soap film is a solution to the Plateau problem. But what happens if a soap film spans a loop of inextensible but flexible wire? We consider this simple query that couples Plateau's problem to Euler's Elastica: a special class of twist-free curves of given length that minimize their total squared curvature energy. The natural marriage of two of the oldest geometrical problems linking physics and mathematics leads to a quest for the shape of a minimal surface bounded by an elastic line: the Euler-Plateau problem. We use a combination of simple physical experiments with soap films that span soft filaments, scaling concepts, exact and asymptotic analysis combined with numerical simulations to explore some of the richness of the shapes that result. Our study raises questions of intrinsic interest in geometry and its natural links to a range of disciplines including materials science, polymer physics, architecture and even art., 14 pages, 4 figures. Supplementary on-line material: http://www.seas.harvard.edu/softmat/Euler-Plateau-problem/

We show that if Gamma is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk Sigma in M with boundary Gamma such that Sigma minimizes the area among the embedded disks with boundary Gamma. Moreover, Sigma is smooth, minimal and embedded everywhere except where the boundary Gamma meets the interior of Sigma. The same result is also valid for homogeneously regular manifolds with sufficiently convex boundary.

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

We prove existence theorems for two-dimensional, noncompact, complete minimal surfaces in ℝn of annular type, which span a given contour and have a finite total curvature end and prescribed asymptotical behavior. For arbitrary rectifiable Jordan curves, we show the existence of such surfaces with a flat end, i.e., within a bounded distance from a 2-plane. For more restricted classes of curves, we prove the existence of minimal surfaces with higher multiplicity flat ends as well as of surfaces with polynomial-type nonflat ends.

It is well known that there exists a global solution to the heat flow of H-systems. If the solution satisfies a certain energy inequality, it is global regular with at most finitely many singularities. Under the same energy inequality, we can show the energy identity of the heat flow of H-systems at finite singular time. The most interesting thing in our proof is that we find the singular points can only occur in the interior of the set in some sense.

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions., 8 pages, 3 figures. To appear in Math. Z

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.

We generalize Meeks and Yau's embeddedness result on the solutions of the Plateau problem to the constant mean curvature disks. We show that any minimizing H-disk in an H_0-convex domain is embedded for any H in [0,H_0). In particular, for the unit ball B in R^3, this implies that for any H in [0,1], any Jordan curve in the unit sphere bounds an embedded H-disk in B., 23 pages, 2 figures, minor changes

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 ) .

We show the area-minimality property of all homogeneous area-minimizing hypercones in Euclidean spaces (classified by Lawlor) following Lawson's original idea in his 72' Trans. A.M.S. paper "The equivariant Plateau problem and interior regularity". Moreover, each of them enjoys (coflat) calibrations singular only at the origin., Improved final version, Acta Mathematica Sinica, English Series, Dec., 2016, Vol. 32

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

Let $\mathbb{R}_{+}^{n+1}$ \ be the half-space model of the hyperbolic space $\mathbb{H}^{n+1}.$ It is proved that if $\Gamma\subset\left\{ x_{n+1}=0\right\} \subset\partial_{\infty}\mathbb{H}^{n+1}$ is a bounded $C^{0}$ Euclidean graph over $\left\{ x_{1}=0,\text{ }x_{n+1}=0\right\} $ then, given $\left\vert H\right\vert, Comment: This is a new version of arXiv:1309.3644 ; Improvements have been made in the proof of the main theorem

The Steiner problem is the problem of finding the shortest network connecting a given set of points. By the singular Plateau Problem, we will mean the problem of finding an area-minimizing surface (or a set of surfaces adjoined so that it is homeomorphic to a 2-complex) spanning a graph. In this paper, we study the parametric versions of the Steiner problem and the singular Plateau problem by a variational method using a modified energy functional for maps. The main results are that the solutions of our one- and two-dimensional variational problems yield length and area minimizing maps respectively, i.e. we provide new methods to solve the Steiner and singular Plateau problems by the use of energy functionals. Furthermore, we show that these solutions satisfy a natural balancing condition along its singular sets. The key issue involved in the two-dimensional problem is the understanding of the moduli space of conformal structures on a 2-complex.

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.

We prove that the [Formula: see text]-solution of the heart equation for the minimal surface with plateau boundary condition is continuous up to boundary in all variables.

In this article, we prove the existence of a minimal annulus bounded by a Jordan curve and a straight line.

In this paper we study a large class of Weingarten surfaces which includes the constant mean curvature one surfaces and flat surfaces in the hyperbolic 3-space. We show that these surfaces can be parametrized by holomorphic data like minimal surfaces in the Euclidean 3-space and we use it to study their completeness. We also establish some existence and uniqueness theorems by studing the Plateau problem at infinity: when is a given curve on the ideal boundary the asymptotic boundary of a complete surface in our family? and, how many embedded solutions are there?

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.

Let \(\) be a parametric variational double integral and γ ⊂ ℝn be a system of several distinct Jordan curves. We prove the existence of multiply connected, conformally parametrized minimizers of \(\) spanned in γ by solving the Douglas problem for parametric functionals on multiply connected schlicht domains. As a by-product we obtain a simple isoperimetric inequality for multiply connected \(\)-minimizers, and we discuss regularity results up to the boundary which follow from corresponding results for the Plateau problem.

The heat flow for the minimal surface under Plateau boundary condition is defined to be a parabolic variational inequality, and then the existence, uniqueness, regularity, continuous dependence on the initial data and the asymptotics are studied. It is applied as a deformation of the level sets in the critical point theory.

We prove that given a convex Jordan curve , the space of properly embedded minimal annuli in the half-space . Moreover, for a fixed positive number , the exterior Plateau problem that consists of finding a properly embedded minimal annulus in the upper half-space, with finite total curvature, boundary has exactly zero, one or two solutions, each one with a different stability character for the Jacobi operator.AMS 2000 Mathematics subject classification: Primary 53A10. Secondary 49Q05; 53C42

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.

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)

Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simple closed curves in the boundary of M which bound unique absolutely area minimizing surfaces in M is dense in the space of simple closed curves in the boundary of M which are nullhomologous in M., 14 pages, 3 figures

We give conditions onH, a continuous and bounded real function inR 3, to obtain at least two solutions for the problem (Dir) below.H can be far from being constant in the sense of [9]. Our motivation is a better understanding of the Plateau problem for the prescribed mean curvature equation.

We show a relation between O'Hara's energy of knots and the Douglas functional. In his paper [4], O'Hara introduced the following energy of knots 2 X JJslxSl {I?p(X) (y)12 sin27r(X y) d for any C2-embedding (o: SI = R/Z 1&3 such that I ('I = 1. This quantity is admitted to be negative, but it is bounded from below (Eknot((o) > -2) and blows up if a knot has a self-intersection. O'Hara showed that there exist only finitely many ambient isotopy classes of knots whose energy and bending energy (see [4] for the definition) are bounded from above by any given constant. In his prize winning paper [3] for the Plateau problem, Douglas employed the Douglas functional D(f) 1 J27t 27 1 ^(e v a) (el-1)12 D ___ k -(evN)-P(V )2da d/i 16Dt = ? sin2(a ,6/2) 7f kop(X) (p(y)12 4 JSXsl sin% t(x-y) dx where (e\v'ia) = (o(a/27r). (See also Struwe [7].) The Douglas functional is closely related to the Dirichlet integral (or more generally the energy functional) of a C1-map f: B2 -+ R3 E(f)=IJIIdf II =I Jj( f 2 of 2)dx dy, 2 jJ2 11tl 2 tJ2 O11x 1 1ay 1 where B2 {z = x+ /Ty E C; lzl < 1} . Indeed D(p) is equal to the energy E(fq,) of the harmonic extension fv, of (0, i.e., Afv, = 0 in B2, f, = ^ on 0B2. (See Douglas [3], Ahlfors [1, Theorems 2-5].) Courant [2] minimized the Dirichlet functional to solve the Plateau problem. The Douglas functional of a curve (0 is an energy of the harmonic surface fv, with the boundary ( . In other Received by the editors July 24, 1991 and, in revised form, August 23, 1991. 1991 Mathematics Subject Classification. Primary 58E12; Secondary 57M25.

CONTENTSIntroduction §1. The Steiner problem and its variations1.1. Fundamental definitions1.2. Globally minimal networks1.3. Locally minimal networks1.4. Closed networks and networks with a fixed boundary1.5. Local structure of minimal networks1.6. Minimal networks in classical ambient spaces1.6.1. The case of the two-dimensional Euclidean plane1.6.2. The case of two-dimensional closed surfaces1.6.3. The case of polyhedra1.6.4. The case of three-dimensional Euclidean space §2. Globally minimal networks on the plane2.1. Minimal spanning trees2.2. Travelling salesperson problem2.3. Minimal Steiner trees2.3.1. Lune property2.3.2. Wedge property2.3.3. Double wedge property2.3.4. Connection between the minimal Steiner tree and the EMST2.3.5. Diamond property2.3.6. Convex hull and Steiner hull2.3.7. The minimal Steiner tree and the Simpson lines2.4. Steiner ratio2.5. Globally minimal networks spanning points lying on a "ladder" and a "zigzag line"2.6. Globally minimal networks spanning points lying on a circle2.7. Improved algorithm for finding a minimal Steiner tree §3. Locally minimal networks on the plane3.1. Complete classification of minimal 2-trees with a convex boundary3.1.1. Rotation number3.1.2. Parquet realization of 2-trees with rotation number not exceeding five3.1.3. Parquets and their properties3.1.4. Classification theorems3.2. Non-degenerate minimal networks with a convex boundary. Cyclic case3.2.1. Trivial networks and their rotation numbers3.2.2. Parquet realization of networks with a convex minimal realization3.2.3. Description of parquets of networks with a convex minimal realization3.3. Minimal 2-trees spanning the vertices of regular n-gons3.4. Convex minimal realization of degenerate Steiner treesReferences

It is shown that hyperbolic 3-space contains an embedded curve γ with the property that any least area disk bounded by γ must intersect P. It follows that least area surfaces in 3-manifolds sometimes intersect more than topologically necessary. Least area surfaces tend to intersect as little as possible, making them useful tools in the study of 3-dimensional manifolds. This paper will present a somewhat surprising phenomenon of excess intersection between least area surfaces in Riemannian 3-manifolds, showing that there is a limit to the generally correct philosophy that "least area surfaces intersect least" [M-Y], [F-H-S]. A plane is said to have least area if any subdisk of the plane is a solution of the Plateau problem for its boundary, so that no disk with the same boundary has less area. We will give an example of an embedded least area plane P and an embedded curve γ lying on one side of P such that any least area disk D which solves the Plateau problem for γ must intersect P. This contrasts with the situation where the disk is embedded; an embedded least area disk is always disjoint from P [M-Y], There is no topological necessity for an intersection between D and P, since by finding a large disk E on P containing DπP and contracting DnP within E, one can push off slightly to obtain a disk bounded by γ and missing P. Thus the example shows that least area surfaces can intersect more than is topologically necessary. The example also shows that the "cut and paste" technique developed by Papakyriakopo ulos [P] and applied by Meeks-Yau [M-Y] in the proofs of the geometric Sphere Theorem and geometric Dehn's Lemma will not extend to the setting of immersed surfaces. THEOREM 1. There is an embedded least area plane P in hyperbolic 3-space and an embedded curve y lying on one side of P such that any least area disk D which solves the Plateau problem for γ must intersect P. Proof. We will present the proof in two steps.

We extend recent results of Guan and Spruck, proving existence results for constant Gaussian curvature hypersurfaces in Hadamard manifolds., Comment: New title. Previously, "Constant Gaussian Curvature Hypersurfaces in Hadamard Manifolds". Mistakes corrected. Introduction revised to reflect recent developments in the field

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.

Let $\alpha$ be a polygonal Jordan curve in $\bfR^3$. We show that if $\alpha$ satisfies certain conditions, then the least-area Douglas-Rad\'{o} disk in $\bfR^3$ with boundary $\alpha$ is unique and is a smooth graph. As our conditions on $\alpha$ are not included amongst previously known conditions for embeddedness, we are enlarging the set of Jordan curves in $\bfR^3$ which are known to be spanned by an embedded least-area disk. As an application, we consider the conjugate surface construction method for minimal surfaces. With our result we can apply this method to a wider range of complete catenoid-ended minimal surfaces in $\bfR^3$.