  In this paper we study bounds for the first eigenvalue of the Paneitz
operator $P$ and its associated third-order boundary operator $B^3$ on
four-manifolds. We restrict to orientable, simply connected, locally confomally
flat manifolds that have at most two umbilic boundary components. The proof is
based on showing that under the hypotheses of the main theorems, the considered
manifolds are confomally equivalent to canonical models. This equivalence is
proved by showing the injectivity of suitable developing maps. Then the bounds
on the eigenvalues are obtained through explicit computations on the canonical
models and its connections with the classes of manifolds that we are
considering. The fact that $P$ and $B^3$ are conformal in four dimensions is
key in the proof.
