  Morphodynamic equations governing the behaviour of active nematic fluids on
deformable curved surfaces are constructed in the large deformation limit.
Emphasis is placed on the formulation of objective rates that account for
normal deformations whilst ensuring that tangential flows are Eulerian, and the
use of the surface derivative (rather than the covariant derivative) in the
nematic free energy, which elastically couples local order to out-of-plane
bending of the surface. Focusing on surface geometry and its dynamical
interplay with the hydrodynamics, several illustrative instabilities are then
characterised. These include cases where the role of the Scriven-Love number
and its nematic analogue are non-negligible, and where the active nematic
forcing can be characterised by an analogue of the F\"{o}ppl-von-K\'{a}rm\'{a}n
number. For the former, flows and changes to the nematic texture are coupled to
surface geometry by viscous dissipation. This is shown to result in non-trivial
relaxation dynamics for a nematic tube. For the latter, the nematic active
forcing couples to the surface bending terms of the nematic free energy,
resulting in extensile (active ruffling) and contractile (active pearling)
instabilities in the tube shape, as well as active bend instabilities in the
nematic texture. In comparison to the flat case, such bend instabilities now
have a threshold set by the extrinsic curvature of the tube. Finally, we
examine a topological defect located on an almost flat surface and show that
there exists a steady state where a combination of defect elasticity, activity
and non-negligible spin-connection drive a shape change in the surface.
