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Uniqueness of asymptotically conical tangent flows
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Singularities of the mean curvature flow of an embedded surface in R^3 are expected to be modelled on self-shrinkers that are compact, cylindrical, or asymptotically conical. In order to understand the flow before and after the singular time, it is crucial to know the uniqueness of tangent flows at the singularity. In all dimensions, assuming the singularity is multiplicity one, uniqueness in the compact case has been established by the second-named author, and in the cylindrical case by Colding-Minicozzi. We show here the uniqueness of multiplicity-one asymptotically conical tangent flows for mean curvature flow of hypersurfaces. In particular, this implies that when a mean curvature flow has a multiplicity-one conical singularity model, the evolving surface at the singular time has an (isolated) regular conical singularity at the singular point. This should lead to a complete understanding of how to "flow through" such a singularity.
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Cited by 1 Pith paper
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The Riemannian Quantitative Isoperimetric Inequality
On a closed Riemannian manifold, the Euclidean quantitative isoperimetric inequality is false in general, holds for generic metrics with the sharp exponent, and holds for real analytic metrics with a modified sharp exponent.
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