Regularized braneworlds of arbitrary codimension

We consider a thick p-brane embedded in an n-dimensional spacetime possessing radial symmetry in the directions orthogonal to the brane. We first consider a static brane, and find a general fine-tuning relationship between the brane and bulk parameters required for the brane to be flat. We then consider the cosmology of a time-dependent brane in a static bulk, and find the Friedmann equation for the brane scale factor a(t). The singularities that would ordinarily arise when considering arbitrary codimensions are avoided by regularizing the brane, giving it a finite profile in the transverse dimensions. However, since we consider the brane to be a strictly local defect, we find that the transverse dimensions must have infinite volume, and hence gravity cannot be localized on the brane without resorting to some infrared cutoff.