  Boundary value problems for operators of Dirac type arise naturally in
connection with the conformal geometry of surfaces immersed in Euclidean
3--space. Recently such boundary value problems have been successfully applied
to a variety of problems from computer graphics. Here we investigate under
which conditions these boundary value problems are elliptic and self--adjoint.
We show that under certain periodic deformations of the boundary data our
operators exhibit non-trivial spectral flow.
