We construct continuously parametrised families of conformally invariant boundary
operators on densities. These generalise to higher orders the first-order conformal
Robin operator and an analogous third-order operator of Chang–Qing. Our families
include operators of critical order on odd-dimensional boundaries. Combined with
conformal Laplacian power operators, the boundary operators yield conformally
invariant fractional Laplacian pseudodifferential operators on the boundary of a
conformal manifold with boundary. We also find and construct new curvature
quantities associated to our new operator families. These have links to the Branson
-curvature
and include higher-order generalisations of the mean curvature and the
-curvature
of Chang–Qing. In the case of the standard conformal hemisphere, the boundary
operator construction is particularly simple; the resulting operators provide an
elementary construction of families of symmetry breaking intertwinors between the
spherical principal series representations of the conformal group of the equator, as
studied by Juhl and others. We discuss applications of our results and techniques in
the setting of Poincaré–Einstein manifolds and also use our constructions to shed
light on some conjectures of Juhl.
