Generalized Weierstrass-Enneper inducing, conformal immersions, and gravity

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.