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arXiv 2002.109112020-02math.DG

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

Jesús Castro-Infantes

备注：: 24 pages, 10 figures
发表信息：: J. Math. Anal. Appl. 507 (2022), no. 2
DOI：: 10.1016/j.jmaa.2021.125831

英文摘要：

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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