Advertisement

Local non-collapsing of volume for the Lagrangian mean curvature flow

  • Knut SmoczykEmail author
Article

Abstract

We prove an optimal control on the time-dependent measure of a measurable set under a reparametrized Lagrangian mean curvature flow of almost calibrated submanifolds in a Calabi–Yau manifold. Moreover we give a classification of those Lagrangian translating solitons in \({{\mathbb {C}}^{m}}\) that evolve by this reparametrized flow.

Mathematics Subject Classification

53C44 53C21 53C42 

Notes

References

  1. 1.
    Angenent, S.: On the formation of singularities in the curve shortening flow. J. Differ. Geom. 33(3), 601–633 (1991)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature, Differential geometry: partial differential equations on manifolds, Los Angeles, CA, : Proc. Sympos. Pure Math., 54, Amer. Math. Soc. Providence, RI 1993, 175–191 (1990)Google Scholar
  3. 3.
    Joyce, D., Lee, Y.-I., Tsui, M.-P.: Self-similar solutions and translating solitons for Lagrangian mean curvature flow. J. Differ. Geom. 84(1), 127–161 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kunikawa, K.: Non existence of eternal solutions to Lagrangian mean curvature flow. arXiv:1611.03594v1 (2016)
  5. 5.
    Martín, F., Savas-Halilaj, A., Smoczyk, K.: On the topology of translating solitons of the mean curvature flow. Calc. Var. Partial Differ. Equ. 54(3), 2853–2882 (2015)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Neves, A.: Singularities of Lagrangian mean curvature flow: zero-Maslov class case. Invent. Math. 168(3), 449–484 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Neves, A., Tian, G.: Translating solutions to Lagrangian mean curvature flow. Trans. Am. Math. Soc. 365(11), 5655–5680 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications. Mem. Amer. Math. Soc. 822, x+99, 0065–9266 (174, 2005)Google Scholar
  10. 10.
    Savas-Halilaj, A., Smoczyk, K.: Lagrangian mean curvature flow of Whitney spheres. arXiv:1802.06304 (to appear in Geometry and Topology) (2018)
  11. 11.
    Smoczyk, K., Der Lagrangesche mittlere Krümmungsfluss.: 102 p., Leipzig, Univ. Leipzig (Habil.-Schr.), German (2000)Google Scholar
  12. 12.
    Sun, J.: Rigidity results on Lagrangian and symplectic translating solitons. Commun. Math. Stat. 3(1), 63–68 (2015)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Sun, J.: Mean curvature decay in symplectic and lagrangian translating solitons. Geom. Dedicata 172, 207–215 (2014)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Wang, M.-T.: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differ. Geom. 57(2), 301–338 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Yau, S.-T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28(201–228), 0010–3640 (1975)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

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

Personalised recommendations