Non-uniqueness of mean curvature flow
Tang-Kai Lee, MIT
Abstract: Abstract: The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, with applications for geometry and topology. 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 flow after singular times. In the talk of Xinrui Zhao, it was presented that even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. The examples we constructed extend to higher dimensions, complementing the surface examples obtained by Ilmanen and White. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish a generalized avoidance principle. We prove that level set flows satisfy this principle in the absence of non-uniqueness.
