Harmonic Maps and Self-Dual Equations for Immersed Surfaces

TL;DR

This paper explores the relationship between harmonic Gauss maps of immersed surfaces and self-dual equations, revealing new connections with gauge fields, constant mean curvature, and generalized Liouville equations.

Contribution

It establishes a link between harmonic Gauss maps and Hitchin's self-dual equations for surfaces in higher dimensions, extending previous results and introducing new gauge-theoretic descriptions.

Findings

Harmonic Gauss maps imply constant mean curvature surfaces.

Hitchin's self-dual equations are derived from surface geometry.

Generalized Liouville equations relate extrinsic geometry to gauge fields.

Abstract

The immersion of the string world sheet, regarded as a Riemann surface, in $R^3$ and $R^4$ is described by the generalized Gauss map. When the Gauss map is harmonic or equivalently for surfaces of constant mean curvature, we obtain Hitchin's self-dual equations, by using $SO(3)$ and $SO(4)$ gauge fields constructed in our earlier studies. This complements our earlier result that $h\surd g\ =\ 1$ surfaces exhibit Virasaro symmetry. The self-dual system so obtained is compared with self-dual Chern-Simons system and a generalized Liouville equation involving extrinsic geometry is obtained. The immersion in $R^n, \ n>4$ is described by the generalized Gauss map. It is shown that when the Gauss map is harmonic, the mean curvature of the immersed surface is constant. $SO(n)$ gauge fields are constructed from the geometry of the surface and expressed in terms of the Gauss map. It is found Hitchin's self- duality relations for the gauge group $SO(2)\times SO(n-2)$.