Skip to main content
Log in

Backward uniqueness for the inverse mean curvature flow

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

In this paper, we prove a backward uniqueness theorem for solutions to the inverse mean curvature flow on a closed manifold. As a consequence, the isometry group of a solution cannot expand within the lifetime of the solution.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Daskalopoulos, P., Huisken, G.: Inverse mean curvature evolution of entire graph. Calc. Var. Partial Differ. Equ. 61(2), 53 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  2. Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32(1), 299–314 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ho, P.T.: Backwards uniqueness of the Yamabe flow. Differ. Geom. Appl. 62, 184–189 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Huang, H.: Backwards uniqueness of the mean curvature flow. Geom. Dedicata 203, 67–71 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Huisken, G., Ilmanen, T.: Higher regularity of the inverse mean curvature flow. J. Differ. Geom. 80(3), 433–451 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59(3), 353–437 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Lee, M.C., Ma, J.: Uniqueness theorems for non-compact mean curvature flow with possibly unbounded curvatures. Commun. Anal. Geom. 29(6), 1475–1508 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. Z. 205(1), 355–372 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, Z.: A note on the backwards uniqueness of the mean curvature flow. Sci. China Math. 62(9), 1793–1798 (2019)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author was supported by NRF grant funded by MSIT (No. NRF-2020R1A2C1A01005698 and NRF-2021R1A4A1032418).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Juncheol Pyo.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ho, P.T., Pyo, J. Backward uniqueness for the inverse mean curvature flow. Geom Dedicata 217, 41 (2023). https://doi.org/10.1007/s10711-023-00781-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10711-023-00781-3

Keywords

Mathematics Subject Classification

Navigation