Direct calculation from the stress tensor of the lateral surface tension of fluctuating fluid membranes

From the tangential and normal stresses associated with the Helfrich Hamiltonian, we calculate the lateral force per unit length, tau, exerted by a planar, fluctuating membrane, as a function of the membrane tension sigma and bending rigidity kappa. We unveil a confusion in the literature concerning the derivation of tau, and we argue, contrary to the present understanding, that tau should differ from the tensionlike coefficient of the fluctuation spectrum. Nontrivial implications concerning the Laplace pressure in vesicles and its relation with the excess area are discussed.