Shocks preempt continuous curvature divergence in interface motion.

The dichotomy between two approaches to interface motion is illustrated in the context of two-dimensional crystal growth. Analyzing singularity formation based on the curvature of the interface predicts a continuous divergence of curvature in contrast to the discrete loss of orientations predicted when the evolution is described by an equation for the two-vector of the interface. We prove that the formation of a shock in the latter approach preempts continuous curvature divergence predicted in the former approach. The results are broadly applicable to kinematic interface motion problems, and we connect them with experiments reported by Maruyama et al. [Phys. Rev. Lett. 85, 2545 (2000)].