A conservative Allen-Cahn equation was introduced by Sun and Beckermann (2007) by subtracting a mean curvature flow from the diffusion equation. In this specific form it has a hyperbolic tangent equilibrium profile between 0 and 1. The equation has been used for phase field lattice Boltzmann simulations of multiphase flow by e.g. Fakhari and Bolster (2017). The advantages of this equation compared to the Cahn-Hilliard equation are that it is only of second order, and that it conserves the mass of the individual phases well. We have used the conservative Allen-Cahn equation for lattice Boltzmann simulations of multiphase flow in complex geometries in three dimensions, and devised a method for improved accuracy when computing the force terms. Results from this study will be presented. Furthermore, in the absence of advection velocity, and when the mobility M is high, the 1/2 level set will move slowly with a certain dynamics, very much resembling the surface diffusion geometric flow. Although, to our knowledge, no proof of this exists, we will argue from numerical results and analytical considerations that this is the case.