In this paper, we generalize White's regularity and structure theory for mean-convex mean curvature flow [45, 46, 48] to the setting with free boundary. A major new challenge in boundary is derive an a priori bound ratio between norm of second fundamental form curvature. We establish such via maximum principle triple-approximation scheme, which combines ideas from Edelen [9], Haslhofer-Hershkov...

Mean curvature flow is the negative gradient flow of volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. If the hypersurface is in general or generic position, then we explain what singularities can occur un...

In this paper we explicitly derive a level set formulation for mean curvature flow in a Riemannian metric space. This extends the traditional geodesic active contour framework which is based on conformal flows. Curve evolution for image segmentation can be posed as a Riemannian evolution process where the induced metric is related to the local structure tensor. Examples on both synthetic and re...