The network flow is the evolution of a regular network of embedded curves under curve shortening flow in the plane, where it is allowed that at triple points three curves meet under a 120 degree condition. A network is called non-regular if at multiple points more than three embedded curves can meet, without any angle condition but with distinct unit tangents. Studying the singularity formation under the flow of regular networks one expects that at the first singular time a non-regular network forms.
In this course we will present recent work together with Tom Ilmanen and Andre Neves, showing that starting from any non-regular initial network there exists a flow of regular networks. The lectures will cover the following material:
1) Short-time existence and higher interior estimates (based on work of Mantegazza, Novaga and Tortorelli).
2) Singularity formation, generalised self-similar shrinking networks and local regularity.
3) Self-similarly expanding networks and their dynamical stability.
4) Desingularising non-regular initial networks and short-time existence.
5) Towards an evolution through singularities.