Backwards uniqueness of the mean curvature flow

  • Hong HuangEmail author
Original Paper


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.


Mean curvature flow Backwards uniqueness Second fundamental form 

Mathematics Subject Classification




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.


  1. 1.
    Chen, B.-L., Yin, L.: Uniqueness and pseudolocality theorems of the mean curvature flow. Comm. Anal. Geom. 15(3), 435–490 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ecker, K., Huisken, G.: Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105, 547–569 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20(1), 237–266 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II. Wiley, Hoboken (1996)zbMATHGoogle Scholar
  5. 5.
    Kotschwar, B.: Backwards uniqueness of the Ricci flow. Int. Math. Res. Not. 21, 4064–4097 (2010)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Kotschwar, B.: A short proof of backward uniqueness for some geometric evolution equations. Int. J. Math. 27(12), 1650102, 17 (2016)MathSciNetCrossRefzbMATHGoogle 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).

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

Personalised recommendations