Integral formulae for the 2 1 wave equation, Einstein-Weyl geometry and Riemann mappings.

Abstract:

The wave equation in 2+1 dimensions can be solved by integrating a function on the plane over parabolae for flat space, or circles for de Sitter space. By integrating over more more general families of curves, the wave equation in more general Einstein-Weyl geometries can be solved. These geometries can be encoded into a scattering map that is a diffeomorphism of the sphere. The relevant curves in the plane are then constructed as from the boundaries of a family of holomorphic discs in a two-dimensional complex manifold that can be obtained from a 3 dimensional family of conformal |(Riemann) mappings.