Research profile
Works on (Non)collapsing, curvature pinching and singularities.
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Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cylinders.
Under mean curvature flow, a closed, embedded hypersurface becomes singular in finite time. For certain classes of mean-convex mean curvature flows, we show the continuity of the first singular time and the limit set "", with respect to initial data. We employ an Angenent-like neck-pinching argument to force singularities in nearby flows. However, since we cannot prescribe initial data, we combine Andrews' -non-collapsed condition and Colding and Minicozzi's uniqueness of tangent flows to place appropriately sized spheres in the region inside the hypersurface.
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