Skip to main content
SpringerLink
Log in

Surfaces in Sol3 Space Foliated by Circles

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol3. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in totally geodesic planes and we classify all such surfaces under the assumption of minimality or flatness.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Daniel, B., Mira, P.: Existence and uniqueness of constant mean curvature spheres in Sol 3. J. Reine Angew. Math. (2012) doi:10.1515/crelle-2012-0016

  2. Enneper, A.: Über die cyclischen Flächen, Nach. Königl. Ges. d. Wissensch. Göttingen (Math. Phys. Kl.) 243–249 (1866)

  3. Inoguchi J., Lee S.: A Weierstrass type representation for minimal surfaces in Sol. Proc. Am. Math. Soc. 146, 2209–2216 (2008)

    Article  MathSciNet  Google Scholar 

  4. López R.: Cyclic surfaces of constant Gauss curvature. Houston J. Math. 27, 799–805 (2001)

    MathSciNet  MATH  Google Scholar 

  5. López R.: Constant mean curvature surfaces in Sol with non-emtpy boundary. Houston J. Math. 38, 1091–1105 (2012)

    MathSciNet  MATH  Google Scholar 

  6. López, R.: Invariant surfaces in Sol3 with constant mean curvature and their computer graphics. Adv. Geom (2013, to appear)

  7. López R., Munteanu M.I.: Translation minimal surfaces in Sol 3. J. Math. Soc. Jpn. 64, 985–1003 (2012)

    Article  MATH  Google Scholar 

  8. López, R., Munteanu, M.I.: Invariant surfaces in the homogeneous space Sol with constant curvature. Math. Nach (2013, to appear)

  9. Nitsche J.C.C.: Lectures on Minimal Surfaces. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  10. Riemann B.: Über die Flächen vom Kleinsten Inhalt be gegebener Begrenzung. Abh. Königl. Ges. d. Wissensch. Göttingen (Mathema. Cl.) 13, 329–333 (1868)

    Google Scholar 

  11. Souam R., Toubiana E.: Totally umbilic surfaces in homogeneous 3-manifolds. Comm. Math. Helv. 84, 673–704 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Thurston W.: Three-dimensional Geometry and Topology (Princeton Mathematical Series 35). Princeton University Press, Princeton (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Geometría y Topología, Universidad de Granada, 18071, Granada, Spain

    Rafael López

  2. Faculty of Mathematics, “Alexandru Ioan Cuza” University of Iasi, Bd. Carol I n. 11, 700506, Iasi, Romania

    Ana Irina Nistor

Authors
  1. Rafael López
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Irina Nistor
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ana Irina Nistor.

Additional information

Part of this work was presented by the authors as a poster at the Conference PADGE, 27–30 August 2012, Leuven, Belgium.

Rights and permissions

Reprints and permissions

About this article

Cite this article

López, R., Nistor, A.I. Surfaces in Sol3 Space Foliated by Circles. Results. Math. 64, 319–330 (2013). https://doi.org/10.1007/s00025-013-0316-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-013-0316-8

Mathematics Subject Classification (2000)

Keywords

Search

Navigation