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Backwards uniqueness of the mean curvature flow

  • Hong HuangEmail author
Original Paper
  • 8 Downloads

Abstract

In this note we prove the backwards uniqueness of the mean curvature flow for (codimension one) hypersurfaces in a Euclidean space. More precisely, let \(F_t, \widetilde{F}_t{:}M^n \rightarrow \mathbb {R}^{n+1}\) be two complete solutions of the mean curvature flow on \(M^n \times [0,T]\) with bounded second fundamental forms. Suppose \(F_T=\widetilde{F}_T\), then \(F_t=\widetilde{F}_t\) on \(M^n \times [0,T]\). This is an analog of a result of Kotschwar on the Ricci flow.

Keywords

Mean curvature flow Backwards uniqueness Second fundamental form 

Mathematics Subject Classification

53C44 

Notes

Acknowledgements

I was partially supported by NSFC No.10671018 and by Laboratory of Mathematics and Complex Systems, Ministry of Education, at BNU. The first version of this note was posted on the arXiv in 2009. Recently there appeared two papers extending our result above to the higher codimension case, cf. [7] (where the ambient spaces may be certain general Riemannian manifolds) and [8] (where the ambient spaces are Euclidean). I would like to thank the authors of these two papers for their comments on the first version of my note, in particular, thank Man-Chun Lee for pointing out a gap in the argument in it. I would also like to thank the referee for the comments.

References

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijingPeople’s Republic of China

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