We investigate the dynamics of active nematic liquid crystals on deformable membranes, focusing on the interplay between active stress and anisotropic curvature coupling. Using a minimal model, we simulate the coupled evolution of the nematic order parameter and membrane height. We demonstrate a continuous transition from a curvature-dominated regime, where topological defects are trapped by local deformation, to an activity-dominated regime exhibiting active turbulence. A scaling analysis reveals that the critical activity threshold ${\zeta }_c$ scales as ${\alpha }^2/\kappa$, where $\alpha$ and $\kappa$ are the coupling constant and bending stiffness, respectively; this relationship is confirmed by our numerical results. Furthermore, we find that significant correlations between the orientational pattern and membrane geometry persist even in the turbulent regime. Specifically, we identify that “walls” in the director field induce characteristic wave-like curvature profiles, providing a mechanism for dynamic coupling between order and shape. These results offer a physical framework for understanding defect-mediated deformation in nonequilibrium biological membranes.

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• Living matter & active matter
• Theories of collective dynamics & active matter