On Delaunay Ends in the DPW Method

TL;DR

This paper studies constant mean curvature 1 surfaces generated by the DPW method, showing they are asymptotic to Delaunay surfaces and establishing uniform closeness and embeddedness properties based on potential parameters.

Contribution

It proves the existence of uniform neighborhoods where these surfaces approximate Delaunay surfaces and are embedded, extending understanding of their asymptotic behavior.

Findings

Surfaces are asymptotic to Delaunay surfaces.

Existence of uniform neighborhoods where surfaces are close to Delaunay.

Embeddedness is confirmed in the unduloid case.

Abstract

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.