A lower bound for the norm of the second fundamental form of minimal hypersurfaces of \(\mathbb{S}^{n+1}\) (Q1423818)

Let \(\varphi: M^n\to \mathbb{S}^{n+1}\) be a minimal immersion of a compact orientable Riemannian manifold \(M^n\) into a unit Euclidean sphere \(\mathbb{S}^{n+1}\). Let \(A\) be the second fundamental form of \(\varphi\) and \(S=| A|^2\). It is known that \(n\) is an upper bound for the first eigenvalue \(\lambda_1\) of the Laplacian \(\Delta\) of \(M\). In this paper, a lower bound for \(S\) is established, which improves the earlier results under the assumption that \(S\) is constant, including the one of the second author in [J. Geom. Phys. 44, 196--201 (2002; Zbl 1033.53047)]. It is shown that there exist a constant \(k>{n\over n-1}\) such that \(S\geq k^{{n-1\over n}} (n-\lambda_1)\). This is concluded from some inequalities obtained for \(\int_M S|\nabla f|^2\), where \(f\) is a first eigenfunction of the Laplacian of \(M^n\).