Advertisement

On the topology of translating solitons of the mean curvature flow

  • Francisco Martín
  • Andreas Savas-Halilaj
  • Knut SmoczykEmail author
Article

Abstract

In the present article we obtain classification results and topological obstructions for the existence of translating solitons of the mean curvature flow in euclidean space.

Mathematics Subject Classification

53C44 53C21 53C42 

Notes

Acknowledgments

The authors would like to thank William H. Meeks III and Antonio Ros for fruitful discussions as well as Jesús Pérez for careful reading of the manuscript.

References

  1. 1.
    Alexandrov, A.D.: Uniqueness theorems for surfaces in the large. Vestnik Leningr. Univ. Math. 11, 5–17 (1956)Google Scholar
  2. 2.
    Altschuler, S., Wu, L.-F.: Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. Partial Differ. Equ. 2, 101–111 (1994)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Clutterbuck, J., Schnürer, O., Schulze, F.: Stability of translating solutions to mean curvature flow. Calc. Var. Partial Differ. Equ. 29, 281–293 (2007)zbMATHCrossRefGoogle Scholar
  4. 4.
    Davila, J., del Pino, M., Nguyen, X.-H.: Finite topology self-translating surfaces for the mean curvature flow in \({\mathbb{R}}^{3}\). arXiv:1501.03867. pp. 1–45 (2015)
  5. 5.
    Eschenburg, J.-H.: Maximum principle for hypersurfaces. Manuscr. Math. 64, 55–75 (1989)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Ferus, D.: On the type number of hypersurfaces in spaces of constant curvature. Math. Ann. 187, 310–316 (1970)zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Differ. Geom. 26, 285–314 (1987)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Halldorsson, H.P.: Helicoidal surfaces rotating/translating under the mean curvature flow. Geom. Dedicata 162, 45–65 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)zbMATHGoogle Scholar
  10. 10.
    Hopf, H.: Differential geometry in the large. Lecture Notes in Mathematics, vol. 1000. Springer-Verlag, Berlin (1983)Google Scholar
  11. 11.
    Huisken, G.: Local and global behaviour of hypersurfaces moving by mean curvature. Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990). Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, pp. 175–191 (1993)Google Scholar
  12. 12.
    Hungerbühler, N., Smoczyk, K.: Soliton solutions for the mean curvature flow. Differ. Integral Equ. 13, 1321–1345 (2000)zbMATHGoogle Scholar
  13. 13.
    Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am. Math. Soc. 108(520) (1994)Google Scholar
  14. 14.
    Lawson, H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. (2) 89, 187–197 (1969)zbMATHCrossRefGoogle Scholar
  15. 15.
    Milnor, J.: Morse Theory. Based on lecture notes by M. Spivak and R. Wells. Annals of Mathematics Studies, No. 51, Princeton University Press, Princeton, NJ (1963)Google Scholar
  16. 16.
    Milnor, J.: Topology from the Differentiable Viewpoint. The University Press of Virginia, Charlottesville, VA (1965)zbMATHGoogle Scholar
  17. 17.
    Nadirashvili, N.: Hadamard’s and Calabi–Yau’s conjectures on negatively curved and minimal surfaces. Invent. Math. 126, 457–465 (1996)zbMATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Nguyen, X.-H.: Complete embedded self-translating surfaces under mean curvature flow. J. Geom. Anal. 23, 1379–1426 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Nguyen, X.-H.: Translating tridents. Commun. Partial Differ. Equ. 34, 257–280 (2009)zbMATHCrossRefGoogle Scholar
  20. 20.
    Savas-Halilaj, A., Smoczyk, K.: Bernstein theorems for length and area decreasing minimal maps. Calc. Var. Partial Differ. Equ. 50, 549–577 (2014)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Schoen, R.M.: Uniqueness, symmetry, and embeddedness of minimal surfaces. J. Differ. Geom. 18, 791–809 (1984)MathSciNetGoogle Scholar
  22. 22.
    Shahriyari, L.: Translating graphs by mean curvature flow. Geom. Dedicata 175, 57–64 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Shahriyari, L.: Translating graphs by mean curvature flow. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)-The Johns Hopkins University (2013)Google Scholar
  24. 24.
    Simons, J.: Minimal varieties in riemannian manifolds. Ann. Math. (2) 88, 62–105 (1968)zbMATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Smith, G.: On complete embedded translating solitons of the mean curvature flow that are of finite genus. arXiv:1501.04149. pp. 1–58 (2015)
  26. 26.
    Smoczyk, K.: A relation between mean curvature flow solitons and minimal submanifolds. Math. Nachr. 229, 175–186 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, X.-J.: Convex solutions to the mean curvature flow. Ann. Math. (2) 173, 1185–1239 (2011)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Francisco Martín
    • 1
  • Andreas Savas-Halilaj
    • 2
  • Knut Smoczyk
    • 2
    Email author
  1. 1.Departmento de Geometría y TopologíaUniversidad de GranadaGranadaSpain
  2. 2.Leibniz Universität Hannover, Institut für Differentialgeometrie and Riemann Center for Geometry and PhysicsHannoverGermany

Personalised recommendations