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 nonflat two-dimensional self-shrinkers. This confirms a conjecture of Colding, Ilmanen, Minicozzi, and White in dimension two.
"A topological property of asymptotically conical self-shrinkers of small entropy." Duke Math. J. 166 (3) 403 - 435, 15 February 2017. https://doi.org/10.1215/00127094-3715082