  For any orientable compact surface with boundary, we compute the regularized
determinant of the Dirichlet-to-Neumann (DN) map in terms of particular values
of dynamical zeta functions by using natural uniformizations, one due to
Mazzeo-Taylor, the other to Osgood-Phillips-Sarnak. We also relate in any
dimension the DN map for the Yamabe operator to the scattering operator for a
conformally compact related problem by using uniformization.
