Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space

Transactions of the American Mathematical Society

Volumn 361, Issue 11, 2009, Pages 5737-5752

  Fernández, Isabel ^a   Mira, Pablo ^b  

^a UNIVERSITY OF SEVILLE   (Spain)

^b TECHNICAL UNIVERSITY OF CARTAGENA   (Spain)

Author keywords

Bernstein problem; Heisenberg group; Holomorphic quadratic differential; Minimal graphs

Indexed keywords

EID: 77950653879     PISSN: 00029947     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0002-9947-09-04645-5     Document Type: Article

References (24)

1. 2 × ℝ, Acta Math. 193 (2004), 141-174. MR2134864 (2006h:53003)
2. U. Abresch, H. Rosenberg, Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1-28. MR2195187 (2006h:53004)
3. L.J. Alías, M. Dajczer, H. Rosenberg, The Dirichlet problem for CMC surfaces in Heisenberg space, Calc. Var. Partial Diff. Equations, 30 (2007), 513-522. MR2332426
4. V. Barone-Adesi, F. Serra-Cassano, D. Vittone, The Bernstein problem for intrinsic graphs in the Heisenberg group and calibrations, Calc. Var. Partial Diff. Equations, 30 (2007), 17-49. MR2333095
5. R.L. Bryant, Surfaces of mean curvature one in hyperbolic space, Astérisque, 154-155 (1987), 321-347. MR955072
6. S.Y. Cheng, S.T. Yau, Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces, Ann. of Math. (2) 104 (1976), 407-419. MR0431061 (55:4063)
7. J.H. Cheng, J.F Hwang, A. Malchiodi, P. Yang, Minimal surfaces in pseudohermi-tian geometry, Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (2005), 129-177. MR2165405 (2006f:53008)
8. P. Collin, H. Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs, preprint, 2007.
9. B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), 87-131. MR2296059 (2008a:53058)
10. B. Daniel, The Gauss map of minimal surfaces in the Heisenberg group, preprint, 2006.
11. B. Daniel, L. Hauswirth, Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg space, preprint, 2007.
12. 2 × ℝ, Amer. J. Math. 129 (2007), 1145-1181. MR2343386
13. I. Fernández, P. Mira, A characterization of constant mean curvature surfaces in homogeneous 3-manifolds, Diff. Geom. Appl., 25 (2007), 281-289. MR2330457
14. C. Figueroa, F. Mercuri, R. Pedrosa, Invariant surfaces of the Heisenberg groups, Ann. Mat. Pura Appl. 177 (1999), 173-194. MR1747630 (2000m:53089)
15. J.A. Gálvez, P. Mira, The Cauchy problem for the Liouville equation and Bryant surfaces, Adv. Math., 195 (2005), 456-490. MR2146351 (2006i:53003)
16. N. Garofalo, S. Pauls, The Bernstein problem in the Heisenberg group, preprint, 2003.
17. 2 × ℝ, preprint, 2007.
18. W.Y. Hsiang, W.T. Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces I, Invent. Math. 98 (1989), 39-58. MR1010154 (90h:53078)
19. J. Inoguchi, T. Kumamoto, N. Ohsugi, Y. Suyama, Differential geometry of curves and surfaces in 3-dimensional homogeneous spaces II, Fukuoka Univ. Sci. Reports 30 (2000), 17-47. MR1763761 (2001c:53085)
20. 2 × ℝ, Acta Math. Sinica, 22 (2006), 1603-1612. MR2262416 (2007g:53007)
21. 1, preprint, 2005.
22. 2 × ℝ, to appear in J. Austr. Math. Soc., 2007.
23. T.Y. Wan, Constant mean curvature surface harmonic map and universal Teichmuller space, J. Differential Geom. 35 (1992), 643-657. MR1163452 (94a:58053)
24. T.Y. Wan, T.K. Au, Parabolic constant mean curvature spacelike surfaces, Proc. Amer. Math. Soc. 120 (1994), 559-564. MR1169052 (94d:53017)