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Evolution of contractions by mean curvature flow

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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.

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References

  1. Chau, A., Chen, J., He, W.: Lagrangian mean curvature flow for entire Lipschitz graphs. Calc. Var. Partial Differ. Equ. 44, 199–220 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Guth, L.: Contraction of areas vs. topology of mappings. Geom. Funct. Anal. 23, 1804–1902 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  3. Guth, L.: Homotopy non-trivial maps with small \(k\)-dilation, pp. 1–7. arXiv:0709.1241v1 (2007)

  4. Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)

    MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  6. Nash, J.: The imbedding problem for Riemannian manifolds. Ann. Math. 63(2), 20–63 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  7. Savas-Halilaj, A., Smoczyk, K.: Homotopy of area decreasing maps by mean curvature flow. Adv. Math. 255, 455–473 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Savas-Halilaj, A., Smoczyk, K.: Bernstein theorems for length and area decreasing minimal maps. Calc. Var. Partial Differ. Equ. 50, 549–577 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  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. Smoczyk, K.: Mean curvature flow in higher codimension-Introduction and survey. Global Differential Geometry. Springer Proceedings in Mathematics, vol. 12, pp. 231–274 (2012)

  11. Smoczyk, K.: Long-time existence of the Lagrangian mean curvature flow. Calc. Var. Partial Differ. Equ. 20, 25–46 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tsui, M.-P., Wang, M.-T.: Mean curvature flows and isotopy of maps between spheres. Comm. Pure Appl. Math. 57, 1110–1126 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wang, M.-T.: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148, 525–543 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. White, B.: A local regularity theorem for mean curvature flow. Ann. Math. 161(2), 1487–1519 (2005)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Differentialgeometrie and Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    Andreas Savas-Halilaj & Knut Smoczyk

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  1. Andreas Savas-Halilaj
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  2. Knut Smoczyk
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Correspondence to Andreas Savas-Halilaj.

Additional information

The first author is supported financially by the grant \(E\Sigma \Pi A\): PE1-417.

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Savas-Halilaj, A., Smoczyk, K. Evolution of contractions by mean curvature flow. Math. Ann. 361, 725–740 (2015). https://doi.org/10.1007/s00208-014-1090-y

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  • DOI: https://doi.org/10.1007/s00208-014-1090-y

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