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arXiv · 2002.10911

On the asymptotic Plateau problem in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$

Jesús Castro-Infantes
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Abstract

We prove some non-existence results for the asymptotic Plateau problem of minimal and area minimizing surfaces in the homogeneous space ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$ with isometry group of dimension 4, in terms of their asymptotic boundary. Also, we show that a properly immersed minimal surface in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$ contained between two bounded entire minimal graphs separated by vertical distance less than $\sqrt{1+4τ^2}π$ have multigraphical ends. Finally, we construct simply connected minimal surfaces with finite total curvature which are not graphs and a family of complete embedded minimal surfaces which are non-proper in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$.

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Jesús Castro-Infantes. 2021-08-30. On the asymptotic Plateau problem in ${\widetilde{\mathrm{SL}}_2(\mathbb{R})}$. https://doi.org/10.1016/j.jmaa.2021.125831

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