Complete submanifolds with parallel mean curvature vector in hyperbolic spaces (Q1882437)

Let \(M^n\) be a complete submanifold with parallel mean curvature vector immersed isometrically in the standard hyperbolic space \(\mathbb{H}^{n+p}(- 1)\) and \(R\) the scalar curvature. Suppose that \(| H|> 1\), where \(| H|\) is the length of the mean curvature vector. It is proven that either \(M^n\) is totally umbilical or \[ \text{inf\,}R\geq n(n- 1)(| H|^2- 1)- \alpha^2, \] where \(\alpha\) is an expression depending in the higher codimension case on \(n\), \(| H|\), \(p\) and in the codimension one case only on \(n\), \(| H|\). It is shown that in the least case this provides a better bound than \textit{Y. Shen's} bound in [Chin. Ann. Math., Ser. B 7, 483--487 (1986; Zbl 0633.53083)].