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On the mean curvature evolution of two-convex hypersurfaces

John Head New York University

Differential Geometry mathscidoc:1609.10314

Journal of Differential Geometry, 94, (2), 241-266, 2013
We study the mean curvature evolution of smooth, closed, twoconvex hypersurfaces in Rn+1 for n ≥ 3.Within this framework we effect a reconciliation between the flow with surgeries—recently constructed by Huisken and Sinestrari in [HS3]—and the wellknown weak solution of the level-set flow: we prove that the two solutions agree in an appropriate limit of the surgery parameters and in a precise quantitative sense. Our proof relies on geometric estimates for certain Lp-norms of the mean curvature which are of independent interest even in the setting of classicalmean curvature flow. We additionally show how our construction can be used to pass these estimates to limits and produce regularity results for the weak solution.
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@inproceedings{john2013on,
  title={ON THE MEAN CURVATURE EVOLUTION OF TWO-CONVEX HYPERSURFACES},
  author={John Head},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914075703367769976},
  booktitle={Journal of Differential Geometry},
  volume={94},
  number={2},
  pages={241-266},
  year={2013},
}
John Head. ON THE MEAN CURVATURE EVOLUTION OF TWO-CONVEX HYPERSURFACES. 2013. Vol. 94. In Journal of Differential Geometry. pp.241-266. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160914075703367769976.
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