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: (i) a cylinder terminated by a spherical cap, and (ii) a bump on an otherwise flat surface. Because the Frank elastic energy on a curved surface generically induces a coupling to deviatoric curvature, D (difference between squared principal curvatures), a +1 defect is induced on both surface types. We find that 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.