  We propose and analyze a new stabilized cut finite element method for the
Laplace-Beltrami operator on a closed surface. The new stabilization term
provides control of the full $\mathbb{R}^3$ gradient on the active mesh
consisting of the elements that intersect the surface. Compared to face
stabilization, based on controlling the jumps in the normal gradient across
faces between elements in the active mesh, the full gradient stabilization is
easier to implement and does not significantly increase the number of nonzero
elements in the mass and stiffness matrices. The full gradient stabilization
term may be combined with a variational formulation of the Laplace-Beltrami
operator based on tangential or full gradients and we present a simple and
unified analysis that covers both cases. The full gradient stabilization term
gives rise to a consistency error which, however, is of optimal order for
piecewise linear elements, and we obtain optimal order a priori error estimates
in the energy and $L^2$ norms as well as an optimal bound of the condition
number. Finally, we present detailed numerical examples where we in particular
study the sensitivity of the condition number and error on the stabilization
parameter.
