Weierstrass-type Representation of Weakly Regular Pseudospherical Surfaces in Euclidean Space

Abstract

The paper presents a generalized Weierstrass representation for pseudospherical surfaces in terms of 3x3 matrices, using moving frames and loop group decompositions. The construction of all such surfaces, starting from a given representation, constitutes the subject of a separate, ulterior report. The appendix of the present work gives an elementary proof of the Birkhoff decomposition theorem for "loop-groups" of spectral parameter based on the real line, instead of the unit circle.