The principal components of the surface stress tenser, sigma, in surfactant systems with internal interfaces in the form of infinite periodic minimal surfaces have been determined from general relations derived by Gurkov and Kralchevsky (Colloids Surf, 47, 45, 1990), and by employing a Helfrich kind of second-order expression for the thermodynamic surface tension, gamma, as a function of the mean curvature, H, and the Gaussian curvature, K. For this hypothetical case it is found that the surface stress tenser, sigma, is isotropic with a constant, positive value everywhere on the surface. However, in order to avoid mechanical instability, an extra term, e.g., proportional to K-2 has to be entailed in the expression for gamma causing the surface stress tenser, sigma, to vary somewhat with the position on the surface. Moreover, because of the necessary inclusion of, e.g., a K-2 term, in reality the equilibrium interfacial structure of a bicontinuous microemulsion or an L(3) (sponge) phase will correspond to a slightly distorted infinite periodic minimal surface.