DPW Potentials for Compact Symmetric CMC Surfaces in $\mathbb{S}^3$

TL;DR

This paper constructs DPW potentials for compact symmetric CMC surfaces in the 3-sphere, enabling their reconstruction via the DPW method, and extends the approach to certain symmetric CMC surfaces.

Contribution

It introduces DPW potentials for Lawson surfaces in $ ext{S}^3$ and generalizes the method to symmetric CMC surfaces in the 3-sphere.

Findings

DPW potentials for Lawson surfaces are explicitly constructed.

The method allows reconstruction of minimal immersions in $ ext{S}^3$.

Extension of the approach to symmetric CMC surfaces in $ ext{S}^3$.

Abstract

Inspired by the work of Heller [12], we show that there exists a DPW potential for the Lawson surface $\xi_{k-1, l-1}$ from which it is possible to reconstruct the minimal immersion $f: \xi_{k-1, l-1} \to \mathbb{S}^3$ 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.