Integer defects, flow localization, and bistability on curved active surfaces

Biological surfaces, such as developing epithelial tissues, exhibit in-plane polar or nematic order and can be strongly curved. Recently, integer (+1) topological defects have been identified as morphogenetic hotspots in living systems. Yet, while +1 defects in active matter on flat surfaces are well-understood, the general principles governing curved active defects remain unknown. Here, we study the dynamics of integer defects in an extensile or contractile polar fluid on two types of morphogenetically-relevant substrates : (1) a cylinder terminated by a spherical cap, and (2) a bump on an otherwise flat surface. Because the Frank elastic energy on a curved surface generically induces a coupling to $\textit{deviatoric}$ curvature, $\mathcal{D}$ (difference between squared principal curvatures), a +1 defect is induced on both surface types. We find that $\mathcal{D}$ leads to surprising effects including localization of orientation gradients and active flows, and particularly for contractility, to hysteresis and bistability between quiescent and flowing defect states.