Constant mean curvature surfaces based on fundamental quadrilaterals

TL;DR

This paper presents a method to construct constant mean curvature (CMC) surfaces with symmetries in spherical and Euclidean spaces using fundamental quadrilaterals and a generalized Weierstrass representation.

Contribution

It introduces a novel approach to building symmetric CMC surfaces via fundamental quadrilaterals and geometric flows on potential spaces.

Findings

Constructed new classes of symmetric CMC surfaces in $ ext{S}^3$ and $ ext{R}^3$.

Developed a generalized Weierstrass representation for these surfaces.

Demonstrated the effectiveness of the method through explicit examples.

Abstract

We describe the construction of CMC surfaces with symmetries in $\mathbb S^3$ and $\mathbb R^3$ using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.