Exploring Surfaces through Methods from the Theory of Integrable Systems. lectures on the Bonnet problem

Abstract

A generic surface in Euclidean 3-space is determined uniquely by its metric and curvature. Classification of all special surfaces where this is not the case, i.e. of surfaces possessing isometries which preserve the mean curvature, is known as the Bonnet problem. Regarding the Bonnet problem, we show how analytic methods of the theory of integrable systems -- such as finite-gap integration, isomonodromic deformation, and loop group description -- can be applied for studying global properties of special surfaces. This paper presents the contents of the lectures given at the School on Differential Geometry on 12-30 April 1999 at the Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste.