  We study graphs of positive extrinsic curvature with a non-removable isolated
singularity in 3-dimensional warped product spaces, and describe their behavior
at the singularity in several natural situations. We use Monge-Amp\`ere
equations to give a classification of the surfaces in 3-dimensional space forms
which are embedded around a non-removable isolated singularity and have a
prescribed, real analytic, positive extrinsic curvature function at every
point. Specifically, we prove that this space is in one-to-one correspondence
with the space of regular, analytic, strictly convex Jordan curves in the
2-dimensional sphere $\mathbb{S^2}$.
