Active Nematics on Evolving Fluid Membranes with Arbitrary Shape and Topology

We present a variational and geometric framework for simulating active nematics on deformable fluid membranes with arbitrary shape and topology. Based on the Onsager principle and a generalized Killing operator, we derive a discrete, structure-preserving formulation of the evolving Stokes equations, yielding a stable variational time integrator for fluid membrane dynamics. The membrane evolves via the Stokesian gradient flow of Helfrich bending energy. To capture nematic dynamics on evolving surfaces, we introduce a Laplacian on the complex line bundle for relaxation and use the Lie derivative for advection. This framework enables the study of defect-driven shape changes in biological membranes and offers design insights for synthetic active materials.