Analysis of thresholding for codimension two motion by mean curvature: a gradient-flow approach

The Merriman-Bence-Osher (MBO) scheme, also known as thresholding or diffusion generated motion, is an efficient numerical algorithm for computing mean curvature flow (MCF). It is fairly well understood in the case of hypersurfaces. This paper establishes the first convergence proof of the scheme in codimension two. We concentrate on the case of the curvature motion of a filament (curve) in $\mathbb{R}^3$. Our proof is based on a new generalization of the minimizing movements interpretation for hypersurfaces (Esedoglu-Otto '15) by means of an energy that approximates the Dirichlet energy of the state function. As long as a smooth MCF exists, we establish uniform energy estimates for the approximations away from the smooth solution and prove convergence towards this MCF. The current result which holds in codimension two relies in a very crucial manner on a new sharp monotonicity formula for the thresholding energy. This is an improvement of an earlier approximate version.