# 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