A general formulation of sixth order geometric flows is proposed in this paper. These geometric flows are obtained via complete variation of a general third order geometric energy functional by means of the gradient descent flow approach under the usual L inner product. We solve these geometric flows by the generalized finite difference method. Comparative experiments and results between lower order flows and sixth order flows are presented. We also display the distinct effects of geometric flows induced from different density functions. Using a sixth order geometric flow and several lower order flows, we solve a surface fairing problem and obtain an aesthetic and pleasing surface.