  We develop a stabilized cut finite element method for the convection problem
on a surface based on continuous piecewise linear approximation and gradient
jump stabilization terms. The discrete piecewise linear surface cuts through a
background mesh consisting of tetrahedra in an arbitrary way and the finite
element space consists of piecewise linear continuous functions defined on the
background mesh. The variational form involves integrals on the surface and the
gradient jump stabilization term is defined on the full faces of the
tetrahedra. The stabilization term serves two purposes: first the method is
stabilized and secondly the resulting linear system of equations is
algebraically stable. We establish stability results that are analogous to the
standard meshed flat case and prove $h^{3/2}$ order convergence in the natural
norm associated with the method and that the full gradient enjoys $h^{3/4}$
order of convergence in $L^2$. We also show that the condition number of the
stiffness matrix is bounded by $h^{-2}$. Finally, our results are verified by
numerical examples.
