Abstract

We begin by investigating the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on cones. We show that the cone apex develops an effective topological charge, which in analogy to electrostatics, leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. We then apply this formalism to develop a minimal model of morphogenesis of a surface where the dynamics of the intrinsic geometry is diffusive growth sourced by topological defects. We show that a positive (negative) defect can dynamically generate a cone (hyperbolic cone). We analytically explain features of the growth profile as a function of position and time, and predict that in the presence of a positive defect, a bump forms with height profile h(t) ~ t^(1/2) for early times t. To incorporate the effect of the mean curvature, we exploit the fact that for axisymmetric surfaces, the extrinsic geometry can be deduced entirely by the intrinsic geometry. We find that the resulting stationary geometry, for polar order and small bending modulus, is a deformed football. We apply our framework to the simple animal Hydra.