Surfaces with parallel mean curvature in \(S^{3}\times \mathbb R\) and \(H^{3}\times \mathbb R\) (Q1931923)

The paper under review concerns surfaces with parallel mean curvature (pmc) vector in a product space \(M^n(c)\times \mathbb R\), where \(M^n(c)\) denotes a simply connected manifold with constant sectional curvature. The authors compute the Laplacian of the squared norm of the traceless part \(\pi\) of the second fundamental form \(\sigma\) of a pmc surface in \(M^n(c)\times \mathbb R\) and obtain a Simons-type formula for \(\phi\), which is used to characterize some particular surfaces in \(M^n(c)\times \mathbb R\); c.f. [\textit{M. H. Batista da Silva}, Ann. Inst. Fourier 61, No. 4, 1299--1322 (2011; Zbl 1242.53066)]. {Theorem 1. } Let \(F^2\) be an immersed pmc 2-sphere in \(M^n(c)\times \mathbb R\) such that: (1) \(| T|^2=0\) or \(| T|^2\geq \frac{2}{3}\) and \(|\sigma|^2 \leq c (2-3| T|^2)\) if \(c<0\), (2) \(| T|^2\leq \frac{2}{3}\) and \(|\sigma|^2 \leq c (2-3| T|^2)\) if \(c>0\). Then \(F^2\) is either a minimal surface in a totally umbilical hypersurface of \(M^n (c)\) or a standard sphere in \(M^3(c)\). {Theorem 2. } Let \(F^2\) be an immersed complete nonminimal pmc surface in \(M^3(c)\times \mathbb R\) with \(c>0\). Assume (i) \(| \phi|^2\leq 2| H|^2 +2c - \frac{5c}{2}| T|^2\) and (ii) either (a) \(| T| =0\) or (b) \(| T| ^2 > \frac{2}{3}\) and \(| H|^2 \geq c | T| ^2 (1-| T|^2) / (3| T|^2 - 2)\). Then either (1) \(|\phi|^2=0\) and \(F^2\) is a round sphere in \(M^3(c)\), or (2) \(|\phi|^2=2| H|^2 +2c\) and \(F^2\) is a torus \(S^1(c) \times S^1(\sqrt{1/c-r^2})\), \(r^2\not= 1/2c\), in \(M^3(c)\). In both statements \(T\) stands for the tangent part of the unit vector \(\xi\) tangent to \(\mathbb R\), whereas \(H\) denotes the mean curvature vector of \(F^2\).