Abstract
We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth.
We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known. By obtaining
speed and curvature pinching estimates for the flows by the approximating speeds, independent of the smoothing
parameter, we may pass to the limit to deduce that the flow by the nonsmooth speed converges to a point in finite time
that, under a suitable rescaling, is round in the
Funding source: Australian Research Council
Award Identifier / Grant number: Discovery Project grant DP120100097
Funding statement: The research of the second author was supported by an Australian Postgraduate Award. The research of the first, third, fourth and fifth authors was supported by Discovery Project grant DP120100097 of the Australian Research Council.
Acknowledgements
The authors are grateful to the Institute for Mathematics and its Applications and the School of Mathematics and Applied Statistics at the University of Wollongong for their support. The third author is grateful to Jiakun Liu for useful discussions. The third and fourth authors are grateful for the support of the Mathematical Sciences Center, Tsinghua University, where part of this work was completed. The authors would like to thank the anonymous referee whose comments have led to improvements in the article.
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