Schwarzian Derivatives and Flows of Surfaces

Over the last decades it has been widely recognized that many completely integrable PDE’s from mathematical physics arise naturally in geometry. Their integrable character in the geometric context – usually associated with the presence of a Lax pair and a spectral deformation – is nothing but the flatness condition of a naturally occurring connection. Examples of interest to geometers generally come from the integrability equations of special surfaces in various ambient spaces: among the best known examples is the sinh-Gordon equation describing constant mean curvature tori in 3-space. In fact, most of the classically studied surfaces, such as isothermic surfaces, surfaces of constant curvature and Willmore surfaces, give rise to such completely integrable PDE. A common thread in many of these examples is the appearance of a harmonic map into some symmetric space, which is well-known to admit a Lax representation with spectral parameter. Once the geometric problem is formulated this way algebro-geometric integration techniques give explicit parameterizations of the surfaces in question in terms of theta functions.