Mean curvature flow is a fundamental geometric evolution equation with natural applications in almost every field of science. To study the evolution past singularities, several notions of weak solutions have been introduced over the last decades. The viscosity solution on the one hand is based on a geometric comparison principle. On the other hand, many other concepts are variational in nature as they are inspired by the gradient flow structure of mean curvature flow. In this talk, I will show that these two viewpoints are equivalent in the following sense: (i) any generic level set of the viscosity solution is a variational solution; (ii) any foliation by variational solutions has to be equal to the unique viscosity solution. This also implies the generic uniqueness of variational solutions.
This is join work with Anton Ullrich (CMU).