  Three types of surface tensions can be defined for lipid membranes: the
internal tension, $\sigma$, conjugated to the real membrane area in the
Hamiltonian, the mechanical frame tension, $\tau$, conjugated to the projected
area, and the "fluctuation tension", $r$, obtained from the fluctuation
spectrum of the membrane height.
  We investigate these surface tensions by means of a Monge gauge lattice Monte
Carlo simulation involving the exact, nonlinear, Helfrich Hamiltonian and a
measure correction for excess entropy of the Monge gauge. Our results for the
relation between $\sigma$ and $\tau$ agrees well with the theoretical
prediction of [J.-B. Fournier and C. Barbetta, Phys. Rev. Lett., 2008, 100,
078103] based on a Gaussian approximation. This provides a valuable knowledge
of~$\tau$ in the standard Gaussian models where the tension is controlled by
$\sigma$. However, contrary to the conjecture in the above paper, we find that
$r$ exhibits no significant difference from $\tau$ over more than five decades
of tension. Our results appear to be valid in the thermodynamic limit and are
robust to changing the ensemble in which the membrane area is controlled.
