CONFORMAL STRUCTURES AND HARMONIC FUNCTIONS

Abstract. We study conformal structures in terms of the kernel of the confor-mal Laplacian. Our main goal is to build a common framework to study both conformal classes of Riemannian metrics and degenerate conformal structures, in particular Carnot-Caratheodary structures (as also piecewise-flat conformal structures). We introduce curvatures and torsions associated to sheafs of functions that determine when the sheaf is the kernel of the conformal Laplacian for a Rie-mannian metric and when the metric is conformally flat. A conformal structure on a smooth manifold M is an equivalence class of Rie-mannian metrics on M, where two metrics g and g ′ are equivalent if for some positive smooth function f: M → R+, g ′ = f2g. Our goal here is to give an alternative characterisation of a conformal structure, in terms of the kernel of the conformal Laplacian, and use this to study various notions of conformal geometry in a manner that applies also to more geeneral cases including Carnot-Caratheodary and PL-metrics. We begin by outlining our characterisation