  Inspired by epithelial morphogenesis, we consider a minimal model for the
shaping of a surface driven by $p$-atic topological defects. We show that a
positive (negative) defect can dynamically generate a (hyperbolic) cone whose
shape evolves diffusively, and predict that a defect of charge $+1/p$ leads to
a final semi-cone angle $\beta$ which satisfies the inequality $\sin\beta \ge 1
- \frac{1}{p} + \frac{1}{2p^2}$. By exploiting the fact that for axisymmetric
surfaces, the extrinsic geometry is tightly coupled to the intrinsic geometry,
we further show that the resulting stationary shape of a membrane with
negligible bending modulus and embedded polar order is a deformed lemon with
two defects at antipodal points. Finally, we close by pointing out that our
results may be relevant beyond epithelial morphogenesis in such contexts as
shape transitions in macroscopic closed spheroidal surfaces such as pollen
grains.
