Douglas Problem Research Articles

On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves

For a smooth closed embedded planar curve Γ, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus 𝔤 ≥ 1 having the curve Γ as boundary, without any prescription on the conormal. In case Γ is a circle we prove that do not exist minimizers and that the infimum of the problem equalsβ𝔤− 4π, whereβ𝔤is the energy of the closed minimizing surface of genus 𝔤. We also prove that the same result also holds if Γ is a straight line for the suitable analogously defined minimization problem on asymptotically flat surfaces. Then we study the case in which Γ is compact, 𝔤 = 1 and the competitors are restricted to a suitable class𝒞of varifolds that includes embedded surfaces. We prove that under suitable assumptions minimizers exists in this class of generalized surfaces.

Area minimizing surfaces of bounded genus in metric spaces

AbstractThe 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. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

Essential Normality of Homogenous Quotient Modules Over the Polydisc: Distinguished Variety Case

In the present paper, we investigate the essential normality of quotient modules over the polydisc. Let I be a homogeneous ideal in $$\mathbb {C}[z_1,\ldots ,z_d]$$ , we show that if the homogeneous quotient module $$[I]^\bot $$ of $$H^2(\mathbb D^d)$$ is essentially normal, then $$\dim _{\mathbb {C}}Z(I)\le 1$$ . It is shown that if Z(I) is distinguished, then $$[I]^\bot $$ is $$(1,\infty )$$ -essentially normal, i.e. the $$[S_{z_i}^*, S_{z_j}]$$ ’s are not necessarily trace class operators but indeed belong to the interpolation ideal $$\mathcal L^{(1,\infty )}$$ , see the monograph “Noncommutative Geometry” of Connes. This result leads to the answer to the polydisc version of Arveson–Douglas problem. Moreover, we study the boundary representation of $$[I]^\bot $$ .

Plateau’s problem in Finsler 3-space

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.

The Douglas problem for parametric double integrals

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.

Constant mean curvature surfaces of annular type

We consider large solutions of annular type to the volume constrained Douglas problem. They are conformally immersed H-surfaces. By rescaling we set the volume functional at one while the boundary curves shrink to the origin. We show that the solutions become spherical in a precise manner. Spherical bubbling may fail if the conformality condition is dropped. We also discuss the rotationally symmetric annular solutions to the H-surface equation and consider some illustrative examples.

An a priori estimate for Douglas problem in Riemannian manifolds

This paper is concerned with the Douglas problem in general Riemannian manifolds. For its perturbed problem we establish a uniform a priori estimate under the energy level min {d*,m* +s0} (see Theorem 3.3). This level seems to be the best possible since in case of ℝn it is just the Douglas condition.

Eine globalanalytische Behandlung des Douglas'schen Problems

In this paper we consider the Douglas problem of genus O. Starting point is the global analysis for minimal surfaces of the type of the disc which was developed by A.J. Tromba. The set of all k-connected minimal surfaces of genus O has a product structure, which is a consequence of the variation of the conformal type. The base space is the space of domain parameters and the fibres are the manifolds of k-connected minimal surfaces of constant conformal type (cf.[7]). It is possible to develop a global analysis also for the more general situation considered here with the following results: -The map which assigns to every minimal surface its boundary curve (in the sense of the Douglas problem) is a Fredholm operator. The index depends on the number and the total order of the branch points. -The analysis allows to prove isolatedness and stability results in a relatively simple way.