Uniqueness of mean curvature flow evolution

The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the evolution after singular times. With Alec Payne, we establish an intersection principle comparing two intersecting flows and prove that level set flows satisfy this principle in the absence of non-uniqueness.