Complete stable minimal submanifolds with a parallel unit normal section in a Euclidean space (Q6635195)

This paper studies complete minimal \(n\)-submanifolds \(\Sigma^{n}\) in Euclidean space \(\mathbb{R}^{n+p}\) which admit a parallel unit normal section. A Bernstein-type theorem is proved for such submanifolds under the assumptions that \(\Sigma^{n}\) is stable and that there exists a parallel unit normal section \(\nu\) such that the \(L^{2}\)-norm of \(A_{\nu}\) is finite and \(|A_{\eta}|^{2}\leq c|A_{\nu}|^{2}\) for any unit normal section \(\eta\) orthogonal to \(\nu\) and for any constant \(c <\frac{2}{n(n^2+2)(p-1)}.\) Here, for a normal section \(\eta\), the tensor \(A_{\eta}\) is the second fundamental form of the submanifold: \(A_{\eta}(X,Y)=\langle \nabla_{X}Y, \eta\rangle\).