For any asymptotically conical self-shrinker with entropy less than or equal to that of a cylinder we show that the link of the asymptotic cone must separate the unit sphere into exactly two connected components, both diffeomorphic to the self-shrinker. Combining this with recent work of Brendle, we conclude that the round sphere uniquely minimizes the entropy among all non-flat two-dimensional self-shrinkers. This confirms a conjecture of Colding-Ilmanen-Minicozzi-White in dimension two.

In 1999 J.C. Léger proved that a one-dimensional set with finite total Menger curvature is 1-rectifiable. In this talk we will present a generalisation of this result to sets of arbitrary dimension and co-dimension. We will give a short sketch of the proof and discuss different higher dimensional versions of Menger curvature known from the literature.