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Networks of curves evolving by curvature in the plane

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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 talk 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.

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