Backwards uniqueness of the mean curvature flow

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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Chen, B.-L., Yin, L.: Uniqueness and pseudolocality theorems of the mean curvature flow. Comm. Anal. Geom. 15(3), 435–490 (2007)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Wiley, Hoboken (1996)

    Google Scholar 

  5. 5.

    Kotschwar, B.: Backwards uniqueness of the Ricci flow. Int. Math. Res. Not. 21, 4064–4097 (2010)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Kotschwar, B.: A short proof of backward uniqueness for some geometric evolution equations. Int. J. Math. 27(12), 1650102, 17 (2016)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Lee, M.-C., John Ma, M.S.: Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures. arXiv:1709.00253

  8. 8.

    Zhang, Z.-H.: A note on the backwards uniqueness of mean curvature flow. Sci. China Math. (2018). https://doi.org/10.1007/s11425-017-9231-4

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hong Huang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Huang, H. Backwards uniqueness of the mean curvature flow. Geom Dedicata 203, 67–71 (2019). https://doi.org/10.1007/s10711-019-00424-6

Download citation

Keywords

  • Mean curvature flow
  • Backwards uniqueness
  • Second fundamental form

Mathematics Subject Classification

  • 53C44