An estimate for the mean curvature of complete submanifolds in a tube (Q760034)

Let \(f: M\to N\) be an isometric immersion of an m-dimensional complete Riemannian manifold M, whose scalar curvature is bounded below, into an n-dimensional Riemannian manifold N, whose sectional curvature \(K_ N\) satisfies \(-\infty <\inf K_ N\) and \(K_ N\leq b\). For \(n>p\geq 1\), let P be a p-dimensional embedded submanifold in N and let \(TP^{\perp}\) be the normal bundle of P. We denote by \(\tau\) (P,\(\lambda)\) the tube of radius \(\lambda\) about P in N. When f(M) is contained in a tube \(\tau\) (P,\(\lambda)\) the author gives an estimate for the sup of \(| H|\), the norm of the mean curvature vector field H of the immersion f. This is a natural extension of a result due to the reviewer and \textit{D. Koutroufiotis} [Arch. Math. 40, 82-85 (1983; Zbl 0502.53042)].