DPW Method Research Articles

Opening nodes in the DPW method: Co-planar case

We combine the DPW method and opening nodes to construct embedded surfaces of positive constant mean curvature with Delaunay ends in euclidean space, with no limitation to the genus or number of ends.

DPW potentials for compact symmetric CMC surfaces in [formula omitted

Inspired by the work of Heller (2013), we show that there exists a DPW potential for the Lawson surface ξk−1,l−1 from which it is possible to reconstruct the minimal immersion f:ξk−1,l−1→S3 via the DPW method. Moreover, we extend the result to surfaces immersed in the 3-sphere with constant mean curvature which satisfy a certain symmetric condition.

Gluing Delaunay ends to minimal n-noids using the DPW method

we construct constant mean curvature surfaces in euclidean space by gluing n half Delaunay surfaces to a non-degenerate minimal n-noid, using the DPW method.

On Delaunay ends in the DPW method

We consider constant mean curvature 1 surfaces in $\mathbb{R}^3$ arising via the DPW method from a holomorphic perturbation of the standard Delaunay potential on the punctured disk. Kilian, Rossman and Schmitt have proven that such a surface is asymptotic to a Delaunay surface. We consider families of such potentials parametrised by the necksize of the model Delaunay surface and prove the existence of a uniform disk on which the surfaces are close to the model Delaunay surface and are embedded in the unduloid case.

Construction of constant mean curvature n-noids using the DPW method

Abstract We construct constant mean curvature surfaces in euclidean space with genus zero and n ends asymptotic to Delaunay surfaces using the DPW method.

The DPW method for constant mean curvature surfaces in 3-dimensional Lorentzian spaceforms, with applications to Smyth type surfaces

We give criteria for singularities of spacelike constant mean curvature surfaces in 3-dimensional de Sitter and anti-de Sitter spaces constructed by the DPW method, which is a generalized Weierstrass representation. We also construct some examples of spacelike CMC surfaces, including analogs of Smyth surfaces with singularities, using appropriate models to visualize them.

Construction of discrete constant mean curvature surfaces in Riemannian spaceforms and applications

In this paper we give a construction for discrete constant mean curvature surfaces in Riemannian spaceforms in terms of integrable systems techniques, which we call the discrete DPW method for discrete constant mean curvature surfaces. Using this construction, we give several examples, and analyze singularities of the parallel constant Gaussian curvature surfaces.

Higher genus minimal surfaces in S3 and stable bundles

We consider compact minimal surfaces of genus 2 which are homotopic to an embedding. We prove that such surfaces can be constructed from a globally defined family of meromorphic connections by the DPW method. The poles of the meromorphic connections are at the Weierstrass points of the Riemann surface and are at most quadratic. For the existence proof of the DPW potential, we give a characterization of stable extensions of spin bundles S by its dual in terms of an associated element of . We also show that the family of holomorphic structures associated to a minimal surface of genus in S 3 is generically stable.

ADDING A UNITON VIA THE DPW METHOD

In this paper we describe how the operation of adding a uniton arises via the DPW method of obtaining harmonic maps into compact Riemannian symmetric spaces from certain holomorphic 1-forms. We exploit this point of view to investigate which unitons preserve finite type property of harmonic maps. In particular, we prove that the Gauss bundle of a harmonic map of finite type into a Grassmannian is also of finite type.

Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.

A scatter correction method for Tl-201 images: A Monte Carlo investigation

Results from the application of a modified dual photopeak window scatter correction method to Monte Carlo simulated Tl-201 emission images are presented. In the Monte Carlo investigation, individual simulations are performed for six radiation emissions of Tl-201. For each emission, point sources of Tl-201 are imaged at various locations in a water-filled elliptical tub phantom using three energy windows. The DPW method is applied to each point source image to estimate the scatter distribution. For point source images in both homogeneous and nonhomogeneous attenuating media, the application of this modified version of DPW results in an approximately sixfold reduction in the scatter fraction and an excellent agreement of the shape of the tails between the estimated scatter distribution and the Monte Carlo-simulated truth. This method is also applied to two views of an extended cardiac distribution within an anthropomorphic phantom, again resulting in at least a sixfold improvement between the scatter estimate and the Monte Carlo-simulated true scatter. >