# 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) Department of Mathematics, New York University, 251 Mercer Street, NY 10012, NEW YORK, UNITED STATES

(2) Department of Mathematics, The Johns Hopkins University, 3400 N. Charles St., MD 21218, 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