2,287 research outputs found
Optimal bounds for ancient caloric functions
For any manifold with polynomial volume growth, we show: The dimension of the
space of ancient caloric functions with polynomial growth is bounded by the
degree of growth times the dimension of harmonic functions with the same
growth. As a consequence, we get a sharp bound for the dimension of ancient
caloric functions on any space where Yau's 1974 conjecture about polynomial
growth harmonic functions holds.Comment: A stronger sharp dimension bound is added which is an equality on
Euclidean space. To appear in Duke Math. Journa
The singular set of mean curvature flow with generic singularities
A mean curvature flow starting from a closed embedded hypersurface in
must develop singularities. We show that if the flow has only generic
singularities, then the space-time singular set is contained in finitely many
compact embedded -dimensional Lipschitz submanifolds plus a set of
dimension at most . If the initial hypersurface is mean convex, then all
singularities are generic and the results apply.
In and , we show that for almost all times the evolving
hypersurface is completely smooth and any connected component of the singular
set is entirely contained in a time-slice. For or -convex hypersurfaces
in all dimensions, the same arguments lead to the same conclusion: the flow is
completely smooth at almost all times and connected components of the singular
set are contained in time-slices. A key technical point is a strong
{\emph{parabolic}} Reifenberg property that we show in all dimensions and for
all flows with only generic singularities. We also show that the entire flow
clears out very rapidly after a generic singularity.
These results are essentially optimal
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