We classify local minimizers of ∫σ2+∮H2 among all conformally flat met-rics in the Euclidean (n+1)-ball, n=4 or n=5, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension n+1=4. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank–Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetriza-tion or Obata-type arguments.