  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.
