A loop group method for Demoulin surfaces in the 3-dimensional real projective space

TL;DR

This paper introduces a loop group approach to characterize Demoulin surfaces in real projective space, linking their geometric properties to harmonic maps and flat connections.

Contribution

It develops a novel loop group method to analyze Demoulin surfaces, connecting their Gauss maps to harmonic maps and flat connections in a 6-symmetric space.

Findings

Demoulin surfaces have conformal first-order Gauss maps.

They are characterized by harmonic first-order Gauss maps.

Demoulin surfaces relate to flat connections on trivial bundles.

Abstract

For a surface in the 3-dimensional real projective space, we define a Gauss map, which is a quadric in $\mathbb R^{4}$ and called the first-order Gauss map. It will be shown that the surface is a Demoulin surface if and only if the first-order Gauss map is conformal, and the surface is a projective minimal coincidence surface or a Demoulin surface if and only if the first-order Gauss map is harmonic. Moreover for a Demoulin surface, it will be shown that the first-order Gauss map can be obtained by the natural projection of the Lorentz primitive map into a 6-symmetric space. We also characterize Demoulin surfaces via a family of flat connections on the trivial bundle $\D \times \SL$ over a simply connected domain $\mathbb{D}$ in the Euclidean 2-plane.