Assume you are given a finite configuration $Γ$ of disjoint rectifiable Jordan curves in $\mathbb{R}^n$. The Plateau-Douglas problem asks whether there exists a minimizer of area among all compact surfaces of genus at most $p$ which span $Γ$. While the solution to this problem is well-known, the classical approaches break down if one allows for singular configurations $Γ$ where the curves are potentially non-disjoint or self-intersecting. Our main result solves the Plateau-Douglas problem for such potentially singular configurations. Moreover, our proof works not only in $\mathbb{R}^n$ but in general proper metric spaces. Thus we are also able to extend previously known existence results of Jürgen Jost as well as of the second author together with Stefan Wenger for regular configurations. In particular, existence is new for disjoint configurations of Jordan curves in general complete Riemannian manifolds. A minimal surface of fixed genus $p$ bounding a given configuration $Γ$ need not always exist, even in the most regular settings. Concerning this problem, we also generalize the approach for singular configurations via minimal sequences satisfying conditions of cohesion and adhesion to the setting of metric spaces.
In this paper we give a geometric argument for bounding the diameter of a connected compact surface (with boundary) of arbitrary codimension in Euclidean space in terms of Topping's diameter bound for closed surfaces (without boundary). The obtained estimate is expected to be optimal for minimal surfaces in the sense that optimality follows if the Topping conjecture holds true. Our result directly implies an explicit nonexistence criterion in the classical Plateau-Douglas problem. We exhibit examples of boundary contours to ensure that our criterion is of different type from classical criteria…
Square metrics $F=\frac{(α+β)^2}α$ are a special class of Finsler metrics. It is the rate kind of metric category to be of excellent geometrical properties. In this paper, we discuss the so-called singular square metrics $F=\frac{(bα+β)^2}α$. A characterization for such metrics to be of vanishing Douglas curvature is provided. Moreover, many analytical examples are achieved by using a special kinds of metrical deformations called $β$-deformations.
This paper describes the work of Jesse Douglas on the Plateau problem, work for which he was awarded a Fields Medal in 1936, and considers the contributions Tibor Rado made in the 1930s.
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 a…
Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of two-dimensional singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$, and we characterize those which are Douglasian or locally projectively flat by some equations. It shows that the main class induced is an $m$-Kropina metric plus a linear part on $β$. For this class, the local structure of Douglasian or (in part) projectively flat case is determined, and in particular we show that a Kropina metric is always Douglasian a…
Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$ and characterize those which are respectively Douglasian and locally projectively flat in dimension $n\ge 3$ by some equations. Our study shows that the main class induced is an $m$-Kropina metric plus a linear part on $β$. For this class with $m\ne -1$, the local structure of projectively flat case is determined, and it is proved that a Douglas $m$-Kropina metric must …
We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas problem in Finsler space can be dealt with in the same way. We also provide a simple isoperimetric inequality for minimal surfaces in Finsler spaces.
We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in $\mathbb{R}^n$.
The Plateau-Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. Th…
Plateau's problem is to show the existence of an area minimizing surface with a given boundary, a problem posed by Lagrange in 1760. Experiments conducted by Plateau showed that an area minimizing surface can be obtained in the form of a film of oil stretched on a wire frame, and the problem came to be called Plateau's problem. Special cases have been solved by Douglas, Rado, Besicovitch, Federer and Fleming, and others. Federer and Fleming used the chain complex of integral currents with its continuous boundary operator to solve Plateau's problem for orientable, embedded surfaces. But integra…
We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Hölder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas and Morrey from the setting of Euclidean space and Riemannian manifolds to…
We define a geometric flow that is designed to change surfaces of cylindrical type spanning two disjoint boundary curves into solutions of the Douglas-Plateau problem of finding minimal surfaces with given boundary curves. We prove that also in this new setting and for arbitrary initial data, solutions of the Teichmüller harmonic map flow exist for all times. Furthermore, for solutions for which a three-point-condition does not degenerate as $t\to\infty$, we show convergence along a sequence $t_i\to\infty$ to a critical point of the area given either by a minimal cylinder or by two minimal dis…
We study metrics with positive scalar curvatures in domains with corners and suggest possible extensions of the concept of positive scalar curvature to singular spaces.
Let $X$ be a compact connected strongly pseudoconvex $CR$ manifold of real dimension $2n-1$ in $\mathbb{C}^{N}$. For $n\ge 3$, Yau solved the complex Plateau problem of hypersurface type by checking a bunch of Kohn-Rossi cohomology groups in 1981. In this paper, we generalize Yau's conjecture on some numerical invariant of every isolated surface singularity defined by Yau and the author to any dimension and prove that the conjecture is true for local complete intersection singularities of dimension $n\ge 3$. As a direct application, we solved complex Plateau problem of hypersurface type for an…
We apply Garnier's method to solve the Plateau problem for maximal surfaces in Minkowski 3-space. Our study relies on the improved version we gave of R. Garnier's resolution of the Plateau problem for polygonal boundary curves in Euclidean 3-space. Since in Minkowski space the method does not allow us to avoid the existence of singularities, the appropriate framework is to consider maxfaces -- generalized maximal surfaces without branch points, introduced by M. Umehara and K. Yamada. We prove that any given spacelike polygonal curve in generic position in Minkowski 3-space bounds at least one …
We prove existence and a.e. regularity of an area minimizing soap film with a bound on energy spanning a given Jordan curve in R^3. The energy of a film is defined to be the sum of its surface area and the length of its singular branched set. The class of surfaces over which area is minimized includes images of disks, integral currents, nonorientable surfaces and soap films as observed by Plateau with a bound on energy. Our area minimizing solution is shown to be a smooth surface away from its branched set which is a union of Lipschitz Jordan curves of finite total length.
We extend the Boutet de Monvel Toeplitz index theorem to complex manifold with isolated singularities following the relative $K$-homology theory of Baum, Douglas, and Taylor for manifold with boundary. We apply this index theorem to study the Arveson-Douglas conjecture. Let $\ball^m$ be the unit ball in $\mathbb{C}^m$, and $I$ an ideal in the polynomial algebra $\mathbb{C}[z_1, \cdots, z_m]$. We prove that when the zero variety $Z_I$ is a complete intersection space with only isolated singularities and intersects with the unit sphere $\mathbb{S}^{2m-1}$ transversely, the representations of $\m…
This paper continues the previous studies in two papers of Huang-Yin [HY3-4] on the flattening problem of a CR singular point of real codimension two sitting in a submanifold in ${\mathbb C}^{n+1}$ with $n+1\ge 3$, whose CR points are non-minimal. Partially based on the geometric approach initiated in [HY3] and a formal theory approach used in [HY4], we are able to provide a very general flattening theorem for a non-degenerate CR singular point. As an application, we provide a solution to the local complex Plateau problem and obtain the analyticity of the local hull of holomorphy near a real a…
In this paper we solve the Plateau problem for spacelike surfaces with constant mean curvature in Lorentz-Minkowski three-space $ł^3$ and spanning two circular (axially symmetric) contours in parallel planes. We prove that rotational symmetric surfaces are the only compact spacelike surfaces in $ł^3$ of constant mean curvature bounded by two concentric circles in parallel planes. As conclusion, we characterize spacelike surfaces of revolution with constant mean curvature as the only that either i) are the solutions of the exterior Dirichlet problem for constant boundary data or ii) have an iso…
