Boundary behavior of conformal deformations

We study conformal deformations of the Euclidean metric in the unit ball $\mathbb{B}^{n}$. We assume that the density associated with the deformation satisfies a Harnack inequality and an arbitrary volume growth condition on the isodiametric profile. We establish a Hausdorff (gauge) dimension estimate for the set $E\subset \partial \mathbb{B}^{n}$ where such a deformation mapping can ``blow up''. We also prove a generalization of Gerasch's theorem in our setting.