Abstract: In this paper, we present a novel approach for efficiently evolving meshes using mean-curvature flow. We use a finite-elements hierarchy that supports an efficient multigrid solver for performing the semi-implicit time-stepping. Although expensive to compute, we show that it is possible to track this hierarchy through the process of surface evolution. As a result, we provide a way to efficiently flow the surface through the evolution, without requiring a costly initialization at the beginning of each time-step. Using our approach, we demonstrate a factor of nearly seven-fold improvement over the non-tracking implementation, supporting the evolution of surfaces consisting of 1M triangles at a rate of just a few seconds per update.