Minimal surfaces with limit ends in \(\mathbb H^2\times\mathbb R\) (Q2872165)

\textit{L. Hauswirth} constructed in [Pac. J. Math. 224, No. 1, 91--117 (2006; Zbl 1108.49031)] properly embedded minimal surfaces in \(\mathbb H^2 \times\mathbb R\) of genus zero having infinitely many ends asymptotic to horizontal slices and two limit ends. Recall that a limit end of a noncompact surface is an accumulation point of the set of ends of the surface. In this paper, the author considers such a minimal surface having a prescribed number of limit ends. He proves that for any \(m\geq 1\), there exists a properly embedded minimal surface in \(\mathbb H^2\times\mathbb R\) with genus zero, infinitely many vertical planar ends (i.e. annular ends asymptotic to vertical geodesic planes) and \(m\) limit ends, which is symmetric with respect to a horizontal slice. This minimal surface is in fact given by a vertical bi-graph. The author also shows that there exists a properly embedded minimal surface in \(\mathbb H^2 \times\mathbb R\) with genus zero, infinitely many vertical planar ends and an infinite countable number of limit ends, which is symmetric with respect to a horizontal slice. NEWLINENEWLINENEWLINE\textit{F. Morabito} and the author constructed in [J. Inst. Math. Jussieu 11, No. 2, 333--349 (2012; Zbl 1238.53004)] a \((2k-3)\)-parameter family \(\mathcal{F}_k\) of properly embedded minimal surfaces in \(\mathbb H^2 \times\mathbb R\) with total curvature \(4\pi (1-k)\), genus zero and \(k\) vertical planar ends for any \(k\geq 2\) (cf. [\textit{J. Pyo}, Ann. Global Anal. Geom. 40, No. 2, 167--176 (2011; Zbl 1242.53074)]). The properly embedded minimal surfaces constructed in this paper are obtained by forming limits for \(k\to\infty\) of certain surfaces \(M_k\in\mathcal{F}_k\).