Thin active nematohydrodynamic layers: Asymptotic theories and instabilities.

Many morphogenetic processes in living systems are driven by active stresses within epithelial sheets that undergo simultaneous changes in thickness, curvature, and internal order. Existing theoretical descriptions of active nematics have largely focused on effectively two-dimensional layers of fixed thickness or prescribed geometry, limiting their ability to capture the coupled thickness-shape dynamics that are central to epithelial morphogenesis. Here, starting from a three-dimensional active nematohydrodynamic description, we derive a hierarchy of low-dimensional continuum models for thin active layers with finite thickness that can deform, flow, and reorient. Using an asymptotic long-wavelength theory, we show how activity couples nematic order to thickness variations and surface geometry in both flat and curved films. In flat layers, allowing thickness to evolve qualitatively modifies classical bend and splay instabilities, producing activity-driven modes that are not directionally constrained. In curved geometries, such as cylindrical films relevant to tubular epithelia and cnidarian body columns, thickness and shape instabilities become intrinsically coupled. Notably, we find that in isotropic phases both extensile and contractile activity can induce nematic order when thickness and shape are dynamic-contrasting sharply with active nematics on fixed substrates. Our results suggest experimentally testable mechanisms by which active stresses in epithelia can drive invagination, thickening, and curvature generation during development. More broadly, this framework provides a minimal physical theory linking active nematic order, thickness regulation, and morphogenesis in living tissues.