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Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify.
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T.H. Colding and W.P. Minicozzi II, Uniqueness of blowups and Lojasiewicz inequalities
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T.H. Colding and W.P. Minicozzi II, Dynamics of closed singularities , Annales de l’Institut Fourier, to appear
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D. Ketover, Self-shrinking Platonic solids , arXiv:1602.07271
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X.H. Nguyen, Construction of Complete Embedded Self-Similar Surfaces under Mean Curvature Flow. Part III , Duke Math. J. 163 (2014), no. 11, 2023–2056
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