Selecta Mathematica

, Volume 24, Issue 5, pp 4811–4838 | Cite as

Asymptotic plateau problem in \({\mathbb H}^2\times {\mathbb R}\)

  • Baris CoskunuzerEmail author


We give a fairly complete solution to the asymptotic Plateau Problem for area minimizing surfaces in \({\mathbb H}^2\times {\mathbb R}\). In particular, we identify the collection of Jordan curves in \(\partial _\infty ({\mathbb H}^2\times {\mathbb R})\) which bounds an area minimizing surface in \({\mathbb H}^2\times {\mathbb R}\). Furthermore, we study the similar problem for minimal surfaces, and show that the situation is highly different.


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Part of this research was carried out at MIT during my visit. I would like to thank them for their great hospitality. I would like to thank the referee for very valuable remarks.


  1. 1.
    Anderson, M.: Complete minimal varieties in \(\mathbb{H}^{n}\). Invent. Math. 69, 477–494 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Collin, P., Rosenberg, H.: Construction of harmonic diffeomorphisms and minimal graphs. Ann. Math. 172, 1879–1906 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Coskunuzer, B.: Minimal Surfaces with Arbitrary Topology in \(\mathbb{H}^{2}\times \mathbb{R}\). arXiv:1404.0214
  4. 4.
    Coskunuzer, B.: Asymptotic \(H\)-plateau problem in \(\mathbb{H}^3\). Geom. Topol. 20, 613–627 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Coskunuzer, B., Meeks III, W.H., Tinaglia, G.: Non-properly embedded \(H\)-planes in \(\mathbb{H}^{2}\times \mathbb{R}\). J. Differ. Geom. 105, 405–425 (2017)CrossRefGoogle Scholar
  6. 6.
    Daniel, B.: Isometric immersions into \(\mathbb{S}^n\times \mathbb{R}\) and \(\mathbb{H}^n\times \mathbb{R}\) and applications to minimal surfaces. Trans. Am. Math. Soc. 361(12), 6255–6282 (2009)CrossRefGoogle Scholar
  7. 7.
    Federer, H.: Geometric Measure Theory. Springer, New York (1969)zbMATHGoogle Scholar
  8. 8.
    Ferrer, L., Martin, F., Mazzeo, R., Rodriguez, M.: Properly embedded minimal annuli in \(\mathbb{H}^{2}\times \mathbb{R}\). arXiv:1704.07788
  9. 9.
    Hauswirth, L., Rosenberg, H., Spruck, J.: Infinite boundary value problems for CMC graphs in \(\mathbb{H}^{2}\times \mathbb{R}\) and \(S^2 \times \mathbb{R}\). Am. J. Math. 131, 195–226 (2009)CrossRefGoogle Scholar
  10. 10.
    Kloeckner, B., Mazzeo, R.: On the asymptotic behavior of minimal surfaces in \(\mathbb{H}^{2}\times \mathbb{R}\). Indiana Univ. Math. J. 66, 631–658 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Martin, F., Mazzeo, R., Rodriguez, M.: Minimal surfaces with positive genus and finite total curvature in \(\mathbb{H}^{2}\times \mathbb{R}\). Geom. Topol. 18, 141–177 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Morabito, F., Rodriguez, M.: Saddle towers and minimal k-noids in \(\mathbb{H}^{2}\times \mathbb{R}\). J. Inst. Math. Jussieu 11, 333–349 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Mazet, L., Rodriguez, M.M., Rosenberg, H.: The Dirichlet problem for the minimal surface equation, with possible infinite boundary data, over domains in a Riemannian surface. Proc. Lond. Math. Soc. 102, 985–1023 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Nelli, B., Rosenberg, H.: Minimal surfaces in \({\mathbb{H}}^2\times {\mathbb{R}}\). Bull. Braz. Math. Soc. 33, 263–292 (2002)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Nelli, B., Sa Earp, R., Santos, W., Toubiana, E.: Uniqueness of H-surfaces in \({\mathbb{H}}^2\times {\mathbb{R}}\), \(|H|\le \frac{1}{2}\), with boundary one or two parallel horizontal circles. Ann. Glob. Anal. Geom. 33, 307–321 (2008)CrossRefGoogle Scholar
  16. 16.
    Pyo, J.: New complete embedded minimal surfaces in \({\mathbb{H}}^2\times {\mathbb{R}}\). Ann. Global Anal. Geom. 40, 167–176 (2011)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pyo, J., Rodriguez, M.: Simply-connected minimal surfaces with finite total curvature in \({\mathbb{H}}^2\times {\mathbb{R}}\). IMRN 2014, 2944–2954 (2014)CrossRefGoogle Scholar
  18. 18.
    Rodriguez, M.M., Tinaglia, G.: Non-proper complete minimal surfaces embedded in \({\mathbb{H}}^2\times {\mathbb{R}}\). IMRN 2015, 4322–4334 (2015). CrossRefzbMATHGoogle Scholar
  19. 19.
    Sa Earp, R., Toubiana, E.: An asymptotic theorem for minimal surfaces and existence results for minimal graphs in \({\mathbb{H}}^2\times {\mathbb{R}}\). Math. Ann. 342, 309–331 (2008)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sa Earp, R., Toubiana, E.: Concentration of total curvature of minimal surfaces in \({\mathbb{H}}^2\times {\mathbb{R}}\). Math. Ann. 369, 1599–1621 (2017)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Mathematics DepartmentBoston CollegeChestnut HillUSA

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