Anisotropic motion by mean curvature in the context of Finsler geometry

We study the anisotropic motion of a hypersurface in the context of the geometry of Finsler spaces. This amounts in considering the evolution in relative geometry, where all quantities are referred to the given Finsler metric $\phi$ representing the anisotropy, which we allow to be a function of space. Assuming that the anisotropy is strictly convex and smooth, we prove that the natural evolution law is of the form "velocity = $H_\phi$", where $H_\phi$ is the relative mean curvature vector of the hypersurface. We derive this evolution law using different approches, such as the variational method of Almgren-Taylor-Wang, the Hamilton-Jacobi equation, and the approximation by means of a reaction-diffusion equation.