Hopf’s theorem on surfaces in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb{R}^3}$$\end{document} with constant mean curvature (Hopf in Math Nach 4:232–249, 1950-51) was a turning point in the study of such surfaces. In recent years, Hopf-type theorems appeared in various ambient spaces, (Abresch and Rosenberg in Acta Math 193:141–174, 2004 and Abresch and Rosenberg in Mat Contemp Sociedade Bras Mat 28:283-298, 2005). The simplest case is the study of surfaces with parallel mean curvature vector in \documentclass[12pt]{minimal}
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\begin{document}$${M_k^n \times \mathbb{R}, n \ge 2}$$\end{document} , where \documentclass[12pt]{minimal}
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\begin{document}$${M_k^n}$$\end{document} is a complete, simply-connected Riemannian manifold with constant sectional curvature k ≠ 0. The case n = 2 was solved in Abresch and Rosenberg 2004. Here we describe some new results for arbitrary n.