Advertisement

Bernstein theorems for length and area decreasing minimal maps

  • Andreas Savas-HalilajEmail author
  • Knut Smoczyk
Article

Abstract

In this article we prove Liouville and Bernstein theorems in higher codimension for length and area decreasing maps between two Riemannian manifolds. The proofs are based on a strong elliptic maximum principle for sections in vector bundles, which we also present in this article.

Mathematics Subject Classification (2000)

53C40 53A07 35J47 58J05 

Notes

Acknowledgments

The first author would like to express his gratitude to the Max-Planck Institute for Mathematics in the Sciences for everything that he benefited during the stay at the Institute and in particular to Professor J. Jost for the scientific support. Moreover, the first author would like to thank Dr. B. Hua for many stimulating conversations.

References

  1. 1.
    Ahlfors, L.V.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43, 359–364 (1938)MathSciNetGoogle Scholar
  2. 2.
    Akhiezer, D.N.: Lie group actions in complex analysis, Aspects of Mathematics E27. Friedr. Vieweg & Sohn, Braunschweig (1995)CrossRefGoogle Scholar
  3. 3.
    Andrews, B., Hopper, C.: The Ricci flow in Riemannian geometry—a complete proof of the differentiable 1/4-pinching sphere theorem. In: Lecture Notes in Mathematics, vol. 2011. Springer, Heidelberg (2011)Google Scholar
  4. 4.
    Bernstein, S.: Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus. Math. Z. 26, 551–558 (1927)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Böhm, C., Wilking, B.: Nonnegatively curved manifolds with finite fundamental groups admit metrics with positive Ricci curvature. Geom. Funct. Anal. 17, 665–681 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bombieri, E., de Giorgi, E., Giusti, E.: Minimal cones and the Bernstein theorem. Invent. Math. 7, 243–269 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Brendle, S.: Ricci flow and the sphere theorem. In: Graduate Studies in Mathematics, vol. 111. American Mathematical Society, Providence (2010)Google Scholar
  8. 8.
    Chern, S.-S., Osserman, R.: Complete minimal surfaces in euclidean \(n\)-space. J. d’Analyse Math. 19, 15–34 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Chern, S.-S.: On the curvatures of a piece of hypersurface in euclidean space. Abh. Math. Sem. Univ. Hamburg 29, 77–91 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci flow: techniques and applications. Part II, Mathematical Surveys and Monographs, vol. 144. American Mathematical Society, Providence (2008)Google Scholar
  11. 11.
    Da Lio, F.: Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Commun. Pure Appl. Anal. 3, 395–415 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Ecker, K.: Regularity theory for mean curvature flow. In: Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser, Boston (2004)Google Scholar
  13. 13.
    Eells, J.: Minimal graphs. Manuscr. Math. 28, 101–108 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Evans, L.C.: A strong maximum principle for parabolic systems in a convex set with arbitrary boundary. Proc. Am. Math. Soc. 138, 3179–3185 (2010)CrossRefzbMATHGoogle Scholar
  15. 15.
    Ferus, D.: On the completeness of nullity foliations. Michigan Math. J. 18, 61–64 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Fleming, W.: On the oriented Plateau problem. Rend. Circ. Mat. Palermo 11, 69–90 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)zbMATHGoogle Scholar
  18. 18.
    Hamilton, R.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17, 255–306 (1982)zbMATHGoogle Scholar
  19. 19.
    Hasanis, Th, Savas-Halilaj, A., Vlachos, Th: On the Jacobian of minimal graphs in \(\mathbb{R}^4\). Bull. Lond. Math. Soc. 43, 321–327 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hasanis, Th, Savas-Halilaj, A., Vlachos, Th: Minimal graphs in \(\mathbb{R}^4\) with bounded Jacobians. Proc. Am. Math. Soc. 137, 3463–3471 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Hildebrandt, S., Jost, J., Widman, K.-O.: Harmonic mappings and minimal submanifolds. Invent. Math. 62, 269–298 (1980/1981)Google Scholar
  22. 22.
    Hopf, E.: Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitzungsberichte Akad. Berlin 19, 147–152 (1927)Google Scholar
  23. 23.
    Jost, J., Xin, Y.-L., Yang, L.: The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension, pp. 1–36. arXiv:1109.6394 (2011)Google Scholar
  24. 24.
    Jost, J., Xin, Y.-L., Yang, L.: The Gauss map of entire maps of higher codimension and the Bernstein type theorems. Calc. Var. Partial Differ. Equ. (2013) (on line first)Google Scholar
  25. 25.
    Jost, J., Xin, Y.-L., Yang, L.: The regularity of harmonic maps into spheres and applications to Bernstein problems. J. Differ. Geom. 90, 131–176 (2012)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Lee, K.-H., Lee, Y.-I.: Mean curvature flow of the graphs of maps between compact manifolds. Trans. Am. Math. Soc. 263, 5745–5759 (2011)CrossRefGoogle Scholar
  27. 27.
    Li, G., Salavessa, I.M.C.: Bernstein–Heinz–Chern results in calibrated manifolds. Rev. Mat. Iberoam. 26, 651–692 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. c. Norm. Super. 11, 211–228 (1978)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Savas-Halilaj, A., Smoczyk, K.: Homotopy of area decreasing maps by mean curvature flow, pp. 1–18 arXiv:1302.0748 (2013)Google Scholar
  30. 30.
    Schoen, R.: The role of harmonic mappings in rigidity and deformation problems. Complex geometry (Osaka, 1990), pp. 179–200. Lecture Notes in Pure and Appl. Math., vol. 143. Dekker, New York (1993)Google Scholar
  31. 31.
    Schoen, R., Simon, L., Yau, S.-T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Smoczyk, K., Wang, G., Xin, Y.-L.: Bernstein type theorems with flat normal bundle. Calc. Var. Partial Differ. Equ. 26, 57–67 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Smoczyk, K., Wang, M.-T.: Mean curvature flows of Lagrangians submanifolds with convex potentials. J. Differ. Geom. 62, 243–257 (2002)zbMATHMathSciNetGoogle Scholar
  35. 35.
    Süss, W.: Über Kennzeichnungen der Kugeln und Affinsphären durch Herrn K.-P. Grotemeyer. Arch. Math. (Basel) 3, 311–313 (1952)Google Scholar
  36. 36.
    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
  37. 37.
    Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100, 197–203 (1978)CrossRefzbMATHGoogle Scholar
  38. 38.
    Wang, M.-T.: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148, 525–543 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Wang, M.-T.: On graphic Bernstein type results in higher codimension. Trans. Am. Math. Soc. 355, 265–271 (2003)Google Scholar
  40. 40.
    Wang, M.-T.: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differ. Geom. 57, 301–338 (2001)zbMATHGoogle Scholar
  41. 41.
    Wang, M.-T.: Deforming area preserving diffeomorphism of surfaces by mean curvature flow. Math. Res. Lett. 8, 651–661 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Wang, X.: A remark on strong maximum principle for parabolic and elliptic systems. Proc. Am. Math. Soc. 109, 343–348 (1990)CrossRefzbMATHGoogle Scholar
  43. 43.
    Weinberger, H.F.: Invariant sets for weakly coupled parabolic and elliptic systems. Rend. Mat. (6) 8, 295–310 (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

Personalised recommendations