Geometry and Lorentzian Harmonic Maps

In 1901 Hilbert proved that there are no complete regular surfaces of constant negative curvature immersed in the Euclidean 3-space, so in order to study these surfaces one needs to allow for singularities. In the classical literature this surfaces, known as Pseudospherical Surfaces, have been  studied as solutions of the Sine-Gordon equation, but the most modern approach is using a certain kind of maps called Lorentzian Harmonic Maps, as follows:

There is a correspondence between Pseudospherical Surfaces (which are geometric objects) and the

1. Lorentzian Harmonic Maps (which are more analytic in nature), so the problem of classifying these surfaces reduces to that of finding all the solutions to the Harmonic equation.

The objective of the project is twofold, first one needs to decide if this new method is the appropriate one (whatever that means, this is part of the question) for describing Pseudospherical Surfaces, but the more immediate and more straightforward objective is to be able to describe exactly which surfaces arise with this method and relate geometrical properties of these with (presumably analytical) properties of the correspondent Lorentzian Harmonic Maps.