Schwarzian Derivatives and Flows of Surfaces

Abstract

This paper goes some way in explaining how to construct an integrable hierarchy of flows on the space of conformally immersed tori in n-space. These flows have first occured in mathematical physics -- the Novikov-Veselov and Davey-Stewartson hierarchies -- as kernel dimension preserving deformations of the Dirac operator. Later, using spinorial representations of surfaces, the same flows were interpreted as deformations of surfaces in 3- and 4-space preserving the Willmore energy. This last property suggest that the correct geometric setting for this theory is Moebius invariant surface geometry. We develop this view point in the first part of the paper where we derive the fundamental invariants -- the Schwarzian derivative, the Hopf differential and a normal connection -- of a conformal immersion into n-space together with their integrability equations. To demonstrate the effectivness of our approach we discuss and prove a variety of old and new results from conformal surface theory. In the the second part of the paper we derive the Novikov-Veselov and Davey-Stewartson flows on conformally immersed tori by Moebius invariant geometric deformations. We point out the analogy to a similar derivation of the KdV hierarchy as flows on Schwarzian's of meromorphic functions. Special surface classes, e.g. Willmore surfaces and isothermic surfaces, are preserved by the flows.