Minimal helix surfaces in Nn×ℝ

Article

Abstract

An immersed surface M in Nn×ℝ is a helix if its tangent planes make constant angle with t. We prove that a minimal helix surface M, of arbitrary codimension is flat. If the codimension is one, it is totally geodesic. If the sectional curvature of N is positive, a minimal helix surfaces in Nn×ℝ is not necessarily totally geodesic. When the sectional curvature of N is nonpositive, then M is totally geodesic. In particular, minimal helix surfaces in Euclidean n-space are planes. We also investigate the case when M has parallel mean curvature vector: A complete helix surface with parallel mean curvature vector in Euclidean n-space is a plane or a cylinder of revolution. Finally, we use Eikonal f functions to construct locally any helix surface. In particular every minimal one can be constructed taking f with zero Hessian.

Keywords

Minimal and helix surfaces Constant angle surfaces Eikonal functions 

Mathematics Subject Classification

53C40 53C42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chen, B.Y.: On the surface with parallel mean curvature vector. Indiana Univ. Math. J. 2, 655–666 (1973) CrossRefGoogle Scholar
  2. 2.
    Di Scala, A., Ruiz-Hernández, G.: Helix submanifolds of Euclidean space. Monatshefte Math. 157, 205–215 (2009) MATHCrossRefGoogle Scholar
  3. 3.
    Di Scala, A., Ruiz-Hernández, G.: Higher codimensional Euclidean helix submanifolds. Kodai Math. J. 33, 192–210 (2010) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dillen, F., Munteanu, M.I.: Constant angle surfaces in ℍ2×ℝ. Bull. Braz. Math. Soc. 40(1), 85–97 (2009); MR2496114 MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Dillen, F., Fastenakels, J., Van der Veken, J., Vrancken, L.: Constant angle surfaces in \(\mathbb{S}^{2}\times \mathbb{R}\). Monatshefte Math. 152(2), 89–96 (2007); MR2346426 MATHCrossRefGoogle Scholar
  6. 6.
    Enomoto, K.: Flat surfaces with mean curvature vector of constant length in Euclidean spaces. Proc. Am. Math. Soc. 110, 211–215 (1990) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Espinar, J.M., Rosenberg, H.: Complete constant mean curvature surfaces and Bernstein type theorems in \(\mathbb{M}^{2}\times \mathbb{R}\). J. Differ. Geom. 82(3), 611–628 (2009). MR2534989 MathSciNetMATHGoogle Scholar
  8. 8.
    Espinar, J.M., Rosenberg, H.: Complete constant mean curvature surfaces in homogeneous spaces. arxiv:0903.2439v1
  9. 9.
    Fischer, A.E.: Riemannian maps between Riemannian manifolds. Contemp. Math. 132, 331–366 (1992) Google Scholar
  10. 10.
    Hoffman, D.A., Osserman, R., Schoen, R.: On the Gauss map of complete surfaces of constant mean curvature in ℝ3 and ℝ4. Comment. Math. Helv. 57, 519–531 (1982) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Montaldo, S., Onnis, I.I.: A note on surfaces on ℍ2×ℝ. Boll. Unione Mat. Ital. 8, 939–950 (2007) MathSciNetGoogle Scholar
  12. 12.
    Ruiz-Hernández, G.: Helix, shadow boundary and minimal submanifolds. Ill. J. Math. 52(4), 1385–1397 (2008) MATHGoogle Scholar
  13. 13.
    Tondeur, P.: Foliations on Riemannian Manifolds. Universitext. Springer, Berlin (1988) MATHGoogle Scholar

Copyright information

© Mathematisches Seminar der Universität Hamburg and Springer 2011

Authors and Affiliations

  1. 1.Mathematics DepartmentMITCambridgeUSA

Personalised recommendations