Extension of Two-Dimensional Mean Curvature Flow with Free Boundary

Abstract

Given a mean curvature flow of compact, embedded \(C^{2}\) surfaces satisfying Neumann free boundary condition on a mean convex, smooth support surface in 3-dimensional Euclidean space, we show that it can be extended as long as its mean curvature and perimeter stay uniformly bounded along the flow.

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Acknowledgements

The author is grateful to Peter Sternberg for suggesting the problem, for insightful discussions and for providing helpful comments on this paper.

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Correspondence to Siao-Hao Guo.

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Guo, SH. Extension of Two-Dimensional Mean Curvature Flow with Free Boundary. J Geom Anal 31, 1568–1624 (2021). https://doi.org/10.1007/s12220-019-00316-x

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Keywords

  • Mean curvature flow
  • Free boundary
  • Extension problem