Commentarii Mathematici Helvetici

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Volume 87, Issue 2, 2012, pp. 463–475
DOI: 10.4171/CMH/260

Smooth compactness of self-shrinkers

Tobias H. Colding[1] and William P. Minicozzi II[2]

(1) New York University, USA
(2) The Johns Hopkins University, Baltimore, United States

We prove a smooth compactness theorem for the space of embedded self-shrinkers in $\mathbb{R}^3$. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all singularities and it plays an important role in studying generic mean curvature flow.

Keywords: Geometric flows, mean curvature flow, self-shrinker, singularities

Colding Tobias, Minicozzi II William: Smooth compactness of self-shrinkers. Comment. Math. Helv. 87 (2012), 463-475. doi: 10.4171/CMH/260