Advertisement

Mathematische Annalen

, Volume 361, Issue 3–4, pp 725–740 | Cite as

Evolution of contractions by mean curvature flow

  • Andreas Savas-HalilajEmail author
  • Knut Smoczyk
Article

Abstract

In this article we investigate length decreasing maps \(f:M\rightarrow N\) between Riemannian manifolds \(M\), \(N\) of dimensions \(m\ge 2\) and \(n\), respectively. Assuming that \(M\) is compact and \(N\) is complete such that
$$\begin{aligned} \sec _M>-\sigma \quad \text {and}\quad {\mathrm{Ric }}_M\ge (m-1)\sigma \ge (m-1)\sec _N\ge -\mu , \end{aligned}$$
where \(\sigma \), \(\mu \) are positive constants, we show that the mean curvature flow provides a smooth homotopy of \(f\) into a constant map.

Mathematics Subject Classification

53C44 53C42 57R52 35K55 

References

  1. 1.
    Chau, A., Chen, J., He, W.: Lagrangian mean curvature flow for entire Lipschitz graphs. Calc. Var. Partial Differ. Equ. 44, 199–220 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Guth, L.: Contraction of areas vs. topology of mappings. Geom. Funct. Anal. 23, 1804–1902 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Guth, L.: Homotopy non-trivial maps with small \(k\)-dilation, pp. 1–7. arXiv:0709.1241v1 (2007)
  4. 4.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)zbMATHGoogle Scholar
  5. 5.
    Lee, K.-W., Lee, Y.-I.: Mean curvature flow of the graphs of maps between compact manifolds. Trans. Am. Math. Soc. 363, 5745–5759 (2011)CrossRefzbMATHGoogle Scholar
  6. 6.
    Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63(2), 20–63 (1956)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Savas-Halilaj, A., Smoczyk, K.: Homotopy of area decreasing maps by mean curvature flow. Adv. Math. 255, 455–473 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Savas-Halilaj, A., Smoczyk, K.: Bernstein theorems for length and area decreasing minimal maps. Calc. Var. Partial Differ. Equ. 50, 549–577 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Smoczyk, K., Tsui, M.-P., Wang, M.-T.: Curvature decay estimates of graphical mean curvature flow in higher co-dimensions, pp. 1–17. arXiv:1401.4154 (2014)
  10. 10.
    Smoczyk, K.: Mean curvature flow in higher codimension-Introduction and survey. Global Differential Geometry. Springer Proceedings in Mathematics, vol. 12, pp. 231–274 (2012)Google Scholar
  11. 11.
    Smoczyk, K.: Long-time existence of the Lagrangian mean curvature flow. Calc. Var. Partial Differ. Equ. 20, 25–46 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Tsui, M.-P., Wang, M.-T.: Mean curvature flows and isotopy of maps between spheres. Comm. Pure Appl. Math. 57, 1110–1126 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Wang, M.-T.: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148, 525–543 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    White, B.: A local regularity theorem for mean curvature flow. Ann. Math. 161(2), 1487–1519 (2005)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Institut für Differentialgeometrie and Riemann Center for Geometry and PhysicsLeibniz Universität HannoverHannoverGermany

Personalised recommendations