Conformal submanifold geometry I-III

TL;DR

This series of papers develops a comprehensive conformal submanifold geometry framework, establishing fundamental structures and equations, and applies them to classify and analyze special surfaces and submanifolds with conformal invariance.

Contribution

It introduces the notion of Moebius structures and conformal Cartan geometries, and applies these to characterize and study various special submanifolds in conformal geometry.

Findings

Established equivalence between Moebius structures and conformal Cartan geometries.

Derived Gauss-Codazzi-Ricci equations and a conformal Bonnet theorem.

Unified theory of Moebius-flat submanifolds including Guichard surfaces and conformally flat hypersurfaces.

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.