Sharp entropy bounds for plane curves and dynamics of the curve shortening flow

We prove that a closed immersed plane curve with total curvature $2 \pi m$ has entropy at least $m$ times the entropy of the embedded circle, as long as it generates a type I singularity under the curve shortening flow (CSF). We construct closed immersed plane curves of total curvature $2 \pi m$ whose entropy is less than $m$ times the entropy of the embedded circle. As an application, we extend Colding–Minicozzi’s notion of a generic mean curvature flow to closed immersed plane curves by constructing a piecewise CSF whose only singularities are embedded circles and type II singularities.