Generalized Weierstrass-Enneper inducing, conformal immersions, and gravity

TL;DR

This paper explores the mathematical framework connecting generalized Weierstrass-Enneper inducing of surfaces with 2-D gravity theories, revealing invariances and links to integrable systems and Liouville-Beltrami gravity.

Contribution

It introduces a novel connection between GWE inducing, conformal immersions, and gravity-related actions, highlighting invariance under integrable hierarchies.

Findings

GWE inducing relates to 2-D gravity actions.

Invariance under Veselov-Novikov hierarchy is demonstrated.

Connections to integrable systems and Liouville-Beltrami gravity are established.

Abstract

Basic quantities related to 2-D gravity, such as Polyakov extrinsic action, Nambu-Goto action, geometrical action, and Euler characteristic are studied using generalized Weierstrass-Enneper (GWE) inducing of surfaces. Connection of the GWE inducing with conformal immersion is made and varius aspects of the theory are shown to be invariant under the modified Veselov-Novikov hierarchy of flows. The geometry of certain surfaces is shown to be connected with the dynamics of infinite and finite dimensional integrable systems. Connections to Liouville-Beltrami gravity are indicated.