## Illinois Journal of Mathematics

### Minimal surfaces in $\widetilde{\mathit{PSL}_{2}(\mathbb{R})}$

Rami Younes

#### Abstract

We study minimal graphs in the homogeneous Riemannian 3-manifold $\widetilde{\mathit{PSL}_{2}(\mathbb{R})}$ and we give examples of invariant surfaces. We derive a gradient estimate for solutions of the minimal surface equation in this space and develop the machinery necessary to prove a Jenkins-Serrin type theorem for solutions defined over bounded domains of the hyperbolic plane.

#### Article information

Source
Illinois J. Math., Volume 54, Number 2 (2010), 671-712.

Dates
First available in Project Euclid: 14 October 2011

https://projecteuclid.org/euclid.ijm/1318598677

Digital Object Identifier
doi:10.1215/ijm/1318598677

Mathematical Reviews number (MathSciNet)
MR2846478

Zentralblatt MATH identifier
1235.53064

#### Citation

Younes, Rami. Minimal surfaces in $\widetilde{\mathit{PSL}_{2}(\mathbb{R})}$. Illinois J. Math. 54 (2010), no. 2, 671--712. doi:10.1215/ijm/1318598677. https://projecteuclid.org/euclid.ijm/1318598677

#### References

• U. Abresch and H. Rosenberg, Generalized Hopf differentials, Mat. Contemp. 28 (2005), 1–28.
• F. Bonahon, Geometric structures on 3-manifolds, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 93–164.
• P. Collin and R. Krust, Le problème de Dirichlet pour l'équation des surfaces minimales sur des domaines non bornés, Bull. Soc. Math. France 119 (1991), 443–462.
• P. Collin and H. Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs, Ann. of Math. (2) 172 (2010), 1879–1906.
• B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), 87–131.
• M. Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & applications, Birkhäuser Boston Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty.
• C. B. Figueroa, F. Mercuri and R. H. L. Pedrosa, Invariant surfaces of the Heisenberg groups, Ann. Mat. Pura Appl. 177 (1999), 173–194.
• D. Fischer-Colbrie and R. Schoen, The structure of complete stable minimal surfaces in $3$-manifolds of nonnegative scalar curvature, Comm. Pure Appl. Math. 33 (1980), 199–211.
• H. Jenkins and J. Serrin, Variational problems of minimal surface type. II. Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal. 21 (1966), 321–342.
• H. B. Lawson, Jr., Lectures on minimal submanifolds. Vol. I, 2nd ed., Mathematics Lecture Series, vol. 9, Publish or Perish Inc., Wilmington, Del., 1980.
• C. B. Morrey, Jr., Multiple integrals in the calculus of variations, Classics in Mathematics, Springer-Verlag, Berlin, 2008. Reprint of the 1966 edition.
• B. Nelli and H. Rosenberg, Minimal surfaces in ${\mathbb H}\sp 2\times\mathbb R$, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 263–292.
• H. Rosenberg, Minimal surfaces in ${\mathbb M}\sp 2\times\mathbb R$, Illinois J. Math. 46 (2002), 1177–1195.
• P. Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), 401–487.
• J. Serrin, A priori estimates for solutions of the minimal surface equation, Arch. Rational Mech. Anal. 14 (1963), 376–383.
• J. Spruck, Interior gradient estimates and existence theorems for constant mean curvature graphs in $M \times\mathbb{R}$, Preprint.