Conformal submanifold geometry I-III

Abstract

In Part I, we develop the notions of a Moebius structure and a conformal Cartan geometry, establish an equivalence between them; we use them in Part II to study submanifolds of conformal manifolds in arbitrary dimension and codimension. We obtain Gauss-Codazzi-Ricci equations and a conformal Bonnet theorem characterizing immersed submanifolds of the conformal n-sphere. These methods are applied in Part III to study constrained Willmore surfaces, isothermic surfaces, Guichard surfaces and conformally-flat submanifolds with flat normal bundle, and their spectral deformations, in arbitrary codimension. The high point of these applications is a unified theory of Moebius-flat submanifolds, which include Guichard surfaces and conformally flat hypersurfaces.