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

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