The conformal-to-Einstein equation on M\"obius surfaces

Abstract

On a conformal manifold, it is well known that parallel sections of the standard tractor bundle with non-vanishing scale are in 1-1 correspondence with solutions of the conformal Einstein equation. In 2 dimensions conformal geometry carries no local information but one can remedy this by equipping the surface with a M\"obius structure. The conformal Einstein equation is then a well defined overdetermined system of linear PDEs on the M\"obius surface. It turns out however that an additional term involving curvature in the form of the Cotton-York tensor appears in the prolongation of the conformal Einstein equation in 2-dimensions, in contrast to the higher dimensional setting. We establish a 1-1 correspondence between solutions of the conformal Einstein equation on M\"obius surfaces with non-vanishing scale and parallel sections of the modified standard tractor connection called the prolongation connection. We also discuss the consequence of this in deriving local obstructions, and also obtain (partial) results on the possible dimensions of the kernel space of the conformal-to-Einstein operator on M\"obius surfaces.