The Bianchi–darboux Transform of L-isothermic Surfaces

Certain types of integrable non-linear PDEs (soliton equations) arise in differential geometry as compatibility conditions for the linear equations obeyed by frames adapted to surfaces in higher dimensional manifolds. In a number of situations, the construction of new solutions of the arising PDE relies on the existence of Bäcklund type transformations for the surfaces and on their permutability properties. Well-known examples include pseudo-spherical surfaces, affine minimal surfaces, and isothermic surfaces in Möbius geometry [CT],[BHPP],[CGS],[TT]. The loop group approach to soliton theory, recently developed by Terng–Uhlenbeck and others [TU], explains the uniformity of results in several of these examples.