## Illinois Journal of Mathematics

### Constant mean curvature $k$-noids in homogeneous manifolds

Julia Plehnert

#### Abstract

For each $k\geq2$, we construct a family of surfaces in $\Sigma(\kappa)\times\mathbb{R}$ with constant mean curvature $H\in[0,1/2]$, where $\kappa+4H^{2}\leq0$ and $\Sigma(\kappa)$ is a two-dimensional space form. The surfaces are invariant under $2\pi/k$-rotations about a vertical fiber of $\Sigma(\kappa)\times\mathbb{R}$, have genus zero, and $2k$ ends. Each surface arises as the sister surface of a minimal graph in a homogeneous $3$-manifold. The domain of the graph is non-convex. We use the sisters of a generalization of Jorge–Meeks-$k$-noids in homogeneous $3$-manifolds as barriers in the conjugate Plateau construction.

#### Article information

Source
Illinois J. Math., Volume 58, Number 1 (2014), 233-249.

Dates
First available in Project Euclid: 1 April 2015

https://projecteuclid.org/euclid.ijm/1427897176

Digital Object Identifier
doi:10.1215/ijm/1427897176

Mathematical Reviews number (MathSciNet)
MR3331849

Zentralblatt MATH identifier
1314.53109

#### Citation

Plehnert, Julia. Constant mean curvature $k$-noids in homogeneous manifolds. Illinois J. Math. 58 (2014), no. 1, 233--249. doi:10.1215/ijm/1427897176. https://projecteuclid.org/euclid.ijm/1427897176

#### References

• P. Collin and H. Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs, Ann. of Math. (2) 172 (2010), no. 3, 1879–1906.
• B. Daniel, Isometric immersions into 3-dimensional homogeneous manifolds, Comment. Math. Helv. 82 (2007), no. 1, 87–131.
• B. Daniel and L. Hauswirth, Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 445–470.
• R. Sa Earp, Parabolic and hyperbolic screw motion surfaces in $\Bbb H^{2}\times\Bbb R$, J. Aust. Math. Soc. 85 (2008), no. 1, 113–143.
• K. Große-Brauckmann and R. B. Kusner, Ruled minimal surfaces in homogeneous manifolds and their sisters, preprint, 2009.
• K. Große-Brauckmann and R. B. Kusner, Conjugate Plateau constructions for homogeneous $3$-manifolds, preprint, 2010.
• R. Gulliver and F. D. Lesley, On boundary branch points of minimizing surfaces, Arch. Ration. Mech. Anal. 52 (1973), 20–25.
• J. Hass and P. Scott, The existence of least area surfaces in 3-manifolds, Trans. Amer. Math. Soc. 310 (1988), no. 1, 87–114.
• H. Karcher, Construction of minimal surfaces, Surveys in geometry, University of Tokyo, 1989.
• H. Karcher, Introduction to conjugate Plateau constructions, Global theory of minimal surfaces, Proceedings of the Clay Mathematics Institute 2001 summer school (Berkeley, CA, USA, June 25–July 27, 2001), American Mathematical Society (AMS), Providence, RI, 2005, pp. 137–161.
• H. B. Lawson, Complete minimal surfaces in $\mathbb{S}^3$, Ann. of Math. (2) 2 (1970), no. 92, 335–374.
• F. Morabito and M. M. Rodríguez, Saddle towers and minimal $k$-noids in $\Bbb H^{2} \times \Bbb R$, J. Inst. Math. Jussieu 11 (2012), no. 2, 333–349.
• J. M. Manzano and F. Torralbo, New examples of constant mean curvature surfaces in $\mathbb{S} ^{2}\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, Michigan Math. J. 63 (2014), 701–723; available at arXiv:\arxivurl1104.1259.
• W. H. Meeks III and S.-T. Yau, The existence of embedded minimal surfaces and the problem of uniqueness, Math. Z. 179 (1982), 151–168.
• J. Plehnert, Constant mean curvature surfaces in homogeneous manifolds, Logos Verlag, Berlin, 2012.
• J. Plehnert, Surfaces with constant mean curvature $1/2$ and genus one in $\mathbb{H}^2\times\mathbb{R}$, preprint, 2012; available at arXiv:\arxivurl1212.2796.
• J. Pyo, New complete embedded minimal surfaces in ${\mathbb{H} ^2\times\mathbb{R}}$, Ann. Global Anal. Geom. 40 (2011), no. 2, 167–176.
• P. Scott, The geometries of $3$-manifolds, Bull. Lond. Math. Soc. 15 (1983), 401–487.
• W. P. Thurston, Three-dimensional geometry and topology, Princeton Mathematical Series, vol. 1, Princeton University Press, Princeton, NJ, 1997.
• R. Younes, Minimal surfaces in $\widetilde{\PSL}_2(\mathbb{R})$, Illinois J. Math. 54 (2010), no. 2, 671–712.