An estimate for the sectional curvature of cylindrically bounded submanifolds (Q2839942)

An isometric immersion \(\varphi\) of a Riemannian manifold \(M^m\) into the product \(N^{n-l}\times\mathbb R^l\) is said to be cylindrically bounded if there exists a geodesic ball \(B_N(r)\subset N\) centered at \(p\in N\) with radius \(r>0\), such that \(\varphi(M)\subset B_N(r)\times\mathbb R^l\). The authors extend the Jorge-Koutrofiotis theorem and give sharp sectional curvature estimates for complete immersed cylindrically bounded \(m\)-submanifolds \(\varphi:M^m\to N^{n-l}\times\mathbb R^l,\;n+l\leq 2m-1\), provided that either \(\varphi\) is proper with certain growth of the norm of the second fundamental form, or the scalar curvature of \(M\) has strong quadratic decay, moreover, the restriction on the codimension cannot be relaxed. In the case where \(M^m\) is compact, the radius of the smallest ball of \(N\) containing \(\pi_N(\varphi(M))\) is expressed in terms of the sectional curvatures of \(M\) and \(N\). For hypersurfaces, the growth rate of the norm of the second fundamental form is improved. The results are obtained via an application of a version of the Omori-Yau maximum principle for the Hessian of a Riemannian manifold due to \textit{S. Pigola} et al. [Mem. Am. Math. Soc. 822, 99 p. (2005; Zbl 1075.58017)].