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

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 $\sqrt{1+4{\tau }^2}\pi$ 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 ${\widetilde{\mathrm{SL}}}_2(\mathbb{R})$.

arXiv comment24 pages, 10 figures

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Cited by in corpus (3)

• Conjugate Plateau constructions in product spaces
• Constant mean curvature graphs with prescribed asymptotic values in $\mathbb{E}(-1,\tau )$
• Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field