Illinois Journal of Mathematics

On the Björling problem in a three-dimensional Lie group

Francesco Mercuri and Irene I. Onnis

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We prove existence and uniqueness of the solution of the Björling problem for minimal surfaces in a three-dimensional Lie group.

Article information

Illinois J. Math., Volume 53, Number 2 (2009), 431-440.

First available in Project Euclid: 23 February 2010

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Zentralblatt MATH identifier

Primary: 53C41 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]


Mercuri, Francesco; Onnis, Irene I. On the Björling problem in a three-dimensional Lie group. Illinois J. Math. 53 (2009), no. 2, 431--440. doi:10.1215/ijm/1266934786.

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  • U. Abresch and H. Rosenberg, Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28.
  • M. Bekkar and T. Sari, Surfaces minimales regleés dans l'espace de Heisenberg $\mathbb{H}_{3}$, Rend. Sem. Sci. Univ. Pol. Torino 50 (1992), 243–254.
  • U. Dierkers, S. Hildebrandt, A. Küster and O. Wohlrab, Minimal surfaces I, Spinger Verlag, Berlin, 1992.
  • I. Fernandez and P. Mira, Holomorphic quadratic differentials and the Bernstein problem in the Heisenberg group, preprint, 2007.
  • C. B. Figueroa, Geometria das subvariedades do grupo de Heisenberg, Ph.D. thesis, University of Campinas, 1996.
  • C. B. Figueroa, F. Mercuri and R. H. L. Pedrosa, Invariant surfaces of the Heisenberg groups, Ann. Mat. Pura Appl. 177 (1999), 173–194.
  • M. Kokubu, Weierstrass representation for minimal surfaces in hyperbolic space, Tôhoku Math. J. 49 (1997), 367–377.
  • F. Mercuri, S. Montaldo and P. Piu, A Weierstrass representation formula for minimal surfaces in $\mathbb{H}_3$ and $\mathbb{H}^2\times\mathbb{R}$, Acta Math. Sinica 22 (2006), 1603–1612.
  • I. I. Onnis, Superfícies em certos espaços homogêneos tridimensionais, Ph.D. thesis, University of Campinas, 2005, available at
  • I. G. Petrovsky, Lectures on partial differential equations, Interscience Publishers, New York, 1954.
  • H. A. Schwarz, Gesammelte mathematische abhandlungen, Band I, Springer, Berlin, 1890.