Conformal invariants from nodal sets II. Manifolds with boundary

In this paper, we study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators on manifolds with boundary. We also consider applications to curvature prescription problems on manifolds with boundary. We relate Dirichlet and Neumann eigenvalues and put the results developed here for the Escobar problem into the more general framework of boundary operators of arbitrary order.