Smooth compactness of self-shrinkers

By T. Colding, W. Minicozzi. 2009

119 citations recorded by the source corpus.

We prove a smooth compactness theorem for the space of embedded self-shrinkers in $\RR^3$. 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.

Subfields: Applied Mathematics, Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/35048c4bcd1a758b731ec0b1b1650b58a4f4c425

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In Applied Mathematics

1. M. Cavalcante
2. Niels Moller
3. J. Espinar
4. N. Kapouleas
5. S. Kleene
6. Leili Shahriyari
7. M. Li
8. S. Brendle

In Mechanical Engineering

1. Leili Shahriyari

Citations in the corpus

• Mean curvature self-shrinkers of high genus: Non-compact examples (2011, background): “torus is still open; see [KM]). Note also that several results involving self-shrinkers in some generality have appeared, most prominently a smooth compactness theorem (for closed, xed genus surfaces [CM1]) and a theory of generic singularities of ColdingMinicozzi, including classication of all H0 c”
• Translating graphs by mean curvature flow (2012, background): “Stability of translating graphs Colding and Minicozzi proved self similar shrinkers are minimal surfaces in a certain conformal metric [5, 6].”
• Halfspace type theorems for self‐shrinkers (2014, background): “More generally, an immersed hypersurface Σ has constant weighted mean curvature Hφ(Σ) = H0 (see [4, 5, 10]).”
• Volume growth, eigenvalue and compactness for self-shrinkers (2011, unknown): “There are several other ways to characterize self-shrinkers (see [5] for hypersurfaces, and high codimensional situation is similar): (1) The one-parameter family of submanifolds √ −tM ⊂ R satisfies MCF equations.”