Linear Stability of the self-similarly shrinking lens

When performing a blowup analysis of singularities in 2D multiphase mean curvature flow, one is led to the notion of self-similar shrinker: Networks whose evolution by mean curvature is given by shrinking homotheties. It can be shown that they are critical points of the interface length functional with a Gaussian weight. Furthermore, this weighted length is decreased during the flow. Hence the dynamic stability of the shrinkers can be studied via stability of the weighted length functional, a matter that is complicated by the existence of, generically, four unstable modes arising from dilation, translation, and rotation. In the talk, I will demonstrate how to perform a linear stability analysis of self-similar shrinkers for the example of the lens.