Journal of Differential Geometry

Half-space Theorems for Minimal Surfaces in Nil$_3$ and Sol$_3$

Benôıt Daniel, William H. Meeks, III, and Harold Rosenberg

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Abstract

We prove some half-space theorems for minimal surfaces in the Heisenberg group Nil$_3$ and the Lie group Sol$_3$ endowed with their standard left-invariant Riemannian metrics. If $\mathcal{S}$ is a properly immersed minimal surface in Nil$_3$ that lies on one side of some entire minimal graph $\mathcal{G}$, then $\mathcal{S}$ is the image of $\mathcal{G}$ by a vertical translation. If $\mathcal{S}$ is a properly immersed minimal surface in Sol$_3$ that lies on one side of a special plane $\mathcal{E}^t$ (see the discussion just before Theorem 1.5 for the definition of a special plane inSol$_3$), then $\mathcal{S}$ is the special plane $\mathcal{E}^u$ for some $u\in \mathbb{R}$.

Article information

Source
J. Differential Geom., Volume 88, Number 1 (2011), 41-59.

Dates
First available in Project Euclid: 4 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1317758868

Digital Object Identifier
doi:10.4310/jdg/1317758868

Mathematical Reviews number (MathSciNet)
MR2819755

Zentralblatt MATH identifier
1237.53053

Citation

Daniel, Benôıt; Meeks, William H.; Rosenberg, Harold. Half-space Theorems for Minimal Surfaces in Nil$_3$ and Sol$_3$. J. Differential Geom. 88 (2011), no. 1, 41--59. doi:10.4310/jdg/1317758868. https://projecteuclid.org/euclid.jdg/1317758868


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