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Bernstein theorems for length and area decreasing minimal maps

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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.

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References

  1. Ahlfors, L.V.: An extension of Schwarz’s lemma. Trans. Am. Math. Soc. 43, 359–364 (1938)

    MathSciNet  Google Scholar 

  2. Akhiezer, D.N.: Lie group actions in complex analysis, Aspects of Mathematics E27. Friedr. Vieweg & Sohn, Braunschweig (1995)

    Book  Google Scholar 

  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)

  4. Bernstein, S.: Über ein geometrisches Theorem und seine Anwendung auf die partiellen Differentialgleichungen vom elliptischen Typus. Math. Z. 26, 551–558 (1927)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bombieri, E., de Giorgi, E., Giusti, E.: Minimal cones and the Bernstein theorem. Invent. Math. 7, 243–269 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brendle, S.: Ricci flow and the sphere theorem. In: Graduate Studies in Mathematics, vol. 111. American Mathematical Society, Providence (2010)

  8. Chern, S.-S., Osserman, R.: Complete minimal surfaces in euclidean \(n\)-space. J. d’Analyse Math. 19, 15–34 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chern, S.-S.: On the curvatures of a piece of hypersurface in euclidean space. Abh. Math. Sem. Univ. Hamburg 29, 77–91 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ecker, K.: Regularity theory for mean curvature flow. In: Progress in Nonlinear Differential Equations and their Applications, vol. 57. Birkhäuser, Boston (2004)

  13. Eells, J.: Minimal graphs. Manuscr. Math. 28, 101–108 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  15. Ferus, D.: On the completeness of nullity foliations. Michigan Math. J. 18, 61–64 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  16. Fleming, W.: On the oriented Plateau problem. Rend. Circ. Mat. Palermo 11, 69–90 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hamilton, R.: Four-manifolds with positive curvature operator. J. Differ. Geom. 24, 153–179 (1986)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  21. Hildebrandt, S., Jost, J., Widman, K.-O.: Harmonic mappings and minimal submanifolds. Invent. Math. 62, 269–298 (1980/1981)

    Google Scholar 

  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. 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)

  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)

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  27. Li, G., Salavessa, I.M.C.: Bernstein–Heinz–Chern results in calibrated manifolds. Rev. Mat. Iberoam. 26, 651–692 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sampson, J.H.: Some properties and applications of harmonic mappings. Ann. Sci. c. Norm. Super. 11, 211–228 (1978)

    MATH  MathSciNet  Google Scholar 

  29. Savas-Halilaj, A., Smoczyk, K.: Homotopy of area decreasing maps by mean curvature flow, pp. 1–18 arXiv:1302.0748 (2013)

  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)

  31. Schoen, R., Simon, L., Yau, S.-T.: Curvature estimates for minimal hypersurfaces. Acta Math. 134, 275–288 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  32. Simons, J.: Minimal varieties in Riemannian manifolds. Ann. Math. 88, 62–105 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  33. Smoczyk, K., Wang, G., Xin, Y.-L.: Bernstein type theorems with flat normal bundle. Calc. Var. Partial Differ. Equ. 26, 57–67 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  34. Smoczyk, K., Wang, M.-T.: Mean curvature flows of Lagrangians submanifolds with convex potentials. J. Differ. Geom. 62, 243–257 (2002)

    MATH  MathSciNet  Google Scholar 

  35. Süss, W.: Über Kennzeichnungen der Kugeln und Affinsphären durch Herrn K.-P. Grotemeyer. Arch. Math. (Basel) 3, 311–313 (1952)

  36. 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 

  37. Yau, S.-T.: A general Schwarz lemma for Kähler manifolds. Am. J. Math. 100, 197–203 (1978)

    Article  MATH  Google Scholar 

  38. 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 

  39. Wang, M.-T.: On graphic Bernstein type results in higher codimension. Trans. Am. Math. Soc. 355, 265–271 (2003)

    Google Scholar 

  40. Wang, M.-T.: Mean curvature flow of surfaces in Einstein four-manifolds. J. Differ. Geom. 57, 301–338 (2001)

    MATH  Google Scholar 

  41. Wang, M.-T.: Deforming area preserving diffeomorphism of surfaces by mean curvature flow. Math. Res. Lett. 8, 651–661 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang, X.: A remark on strong maximum principle for parabolic and elliptic systems. Proc. Am. Math. Soc. 109, 343–348 (1990)

    Article  MATH  Google Scholar 

  43. Weinberger, H.F.: Invariant sets for weakly coupled parabolic and elliptic systems. Rend. Mat. (6) 8, 295–310 (1975)

    Google Scholar 

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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.

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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

Authors
  1. Andreas Savas-Halilaj
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  2. Knut Smoczyk
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Corresponding author

Correspondence to Andreas Savas-Halilaj.

Additional information

Communicated by J. Jost.

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

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Savas-Halilaj, A., Smoczyk, K. Bernstein theorems for length and area decreasing minimal maps. Calc. Var. 50, 549–577 (2014). https://doi.org/10.1007/s00526-013-0646-0

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  • DOI: https://doi.org/10.1007/s00526-013-0646-0

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