Abstract
We construct complete, embedded minimal annuli asymptotic to vertical planes in the Riemannian 3-manifold \(\widetilde{\textrm{PSL}}_2({\mathbb {R}},\tau )\). The boundary of these annuli consists of 4 vertical lines at infinity. They are constructed by taking the limit of a sequence of compact minimal annuli. The compactness is obtained from an estimate of curvature which uses foliations by minimal surfaces. This estimate is independent of the index of the surface. We also prove the existence of a one-periodic family of Riemann’s type examples. The difficulty of the construction comes from the lack of symmetry of the ambient space \(\widetilde{\textrm{PSL}}_2({\mathbb {R}},\tau )\).
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Collin, P., Hauswirth, L. & Nguyen, M.H. Construction of minimal annuli in \(\widetilde{\textrm{PSL}}_2({\mathbb {R}},\tau )\) via a variational method. Math. Ann. 387, 2105–2155 (2023). https://doi.org/10.1007/s00208-022-02502-9
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DOI: https://doi.org/10.1007/s00208-022-02502-9