Local obstructions to a conformally invariant equation on M\"obius surfaces

TL;DR

This paper investigates a conformally invariant overdetermined PDE system on M"obius surfaces, deriving local algebraic obstructions and conditions for the existence of solutions based on conformal invariants.

Contribution

It introduces local algebraic obstructions for the scalar-flat M"obius Einstein-Weyl equation, expanding understanding of solution conditions on M"obius surfaces.

Findings

Obstructions are given by resultants of polynomial equations.

Vanishing of resultants is necessary for solutions.

Provides conditions based on conformal invariants.

Abstract

On a M\"obius surface, as defined by D. Calderbank, we study a variant of the Einstein-Weyl (EW) equation which we call scalar-flat M\"obius EW (sf-MEW). This is a conformally invariant, finite type, overdetermined system of semi-linear partial differential equations. We derive local algebraic constraints for this equation to admit a solution and give local obstructions. In the generic case when a certain invariant of the M\"obius structure given by a symmetric tensor $M_{ab}$ is non-zero, the obstructions are given by resultants of 3 polynomial equations whose coefficients are conformal invariants of the M\"obius structure. The vanishing of the resultants is a necessary condition for there to be solutions to sf-MEW.