John Ryan University of Arkansas

Conformal Laplace and Dirac Operators on Spheres and Hyperbolas.

In order to solve Dirichlet, Neumann and other boundary value problems for domains on spheres and hyperbolas and other conformally flat manifolds one can set up Dirac operators and Hardy spaces for these domains. Using this machinery we may introduce a Laplacian on spheres and hyperbolas that are conformally equivalent to the Laplacian in Euclidean space. This Laplacian factors as D(D+x)=(D-x)D, where D is the Dirac operator, x is a point varying on the sphere/hyperbola and multiplication is defined using a Clifford algebra. The two seperate factorizations of these Laplacians give rise to two seperate Green's formulae for solutions to the Laplace equations. A Poisson kernel is introduced to solve the Dirichlet problem on L^p spaces for 1