Illinois Journal of Mathematics

Constant mean curvature $k$-noids in homogeneous manifolds

Julia Plehnert

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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

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1427897176

Digital Object Identifier
doi:10.1215/ijm/1427897176

Mathematical Reviews number (MathSciNet)
MR3331849

Zentralblatt MATH identifier
1314.53109

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15]

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


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