Tangential Tensor Fields on Deformable Surfaces -- How to Derive Consistent $L^2$-Gradient Flows

We consider gradient flows of surface energies which depend on the surface by a parameterization and on a tangential tensor field. The flow allows for dissipation by evolving the parameterization and the tensor field simultaneously. This requires the choice of a notation for independence. We introduce different gauges of surface independence and show their consequences for the evolution. In order to guarantee a decrease in energy, the gauge of surface independence and the time derivative have to be chosen consistently. We demonstrate the results for a surface Frank-Oseen-Hilfrich energy.