204 research outputs found
The mean curvature at the first singular time of the mean curvature flow
Consider a family of smooth immersions
of closed hypersurfaces in moving by the mean curvature flow
, for .
We prove that the mean curvature blows up at the first singular time if all
singularities are of type I. In the case , regardless of the type of a
possibly forming singularity, we show that at the first singular time the mean
curvature necessarily blows up provided that either the Multiplicity One
Conjecture holds or the Gaussian density is less than two. We also establish
and give several applications of a local regularity theorem which is a
parabolic analogue of Choi-Schoen estimate for minimal submanifolds
Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space
The rigidity of the Positive Mass Theorem states that the only complete
asymptotically flat manifold of nonnegative scalar curvature and zero mass is
Euclidean space. We study the stability of this statement for spaces that can
be realized as graphical hypersurfaces in Euclidean space. We prove (under
certain technical hypotheses) that if a sequence of complete asymptotically
flat graphs of nonnegative scalar curvature has mass approaching zero, then the
sequence must converge to Euclidean space in the pointed intrinsic flat sense.
The appendix includes a new Gromov-Hausdorff and intrinsic flat compactness
theorem for sequences of metric spaces with uniform Lipschitz bounds on their
metrics.Comment: 31 pages, 2 figures, v2: to appear in Crelle's Journal, many minor
changes, one new exampl
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