Applications of Bochner formula to minimal submanifold of the sphere.
Let \(M\) be an \(n\)-dimensional closed smooth manifold, and let \(\psi:M \to S^{n+p}\) be a minimal immersion into the \((n+p)\)-dimensional sphere of constant sectional curvature one. Denote by \(B, S\) the second fundamental form of \(\psi\) and the squared length of \(B\), respectively. Denote by \(\lambda_ 1\) the first eigenvalue of the Laplacian of \(M\) in the induced metric by \(\psi\). The author gives a new estimate of \(S\) for \(p=1\): Let \(f\) be an eigenfunction belonging to \(\lambda_ 1\), and \(k\) be the dimension of the kernel of \(B\). Then, it is shown that \[ \int_ M S| \nabla f| ^ 2 \geq ((n-k)(n-1)/n(n-k-1))(n-\lambda_ 1) \int_ M | \nabla f| ^ 2. \] In particular, if \(S\) is constant, then \(S\geq ((n-k)(n-1)/n(n-k-1))(n-\lambda_ 1)\). This estimate is an improvement for hypersurfaces of a former result proved by \textit{P.-F. Leung} [Math. Z. 183, 75--86 (1983; Zbl 0491.53045)], namely, if \(S\) is constant, then \(S \geq n-\lambda_ 1\) holds for general \(p\). Moreover, some estimates of \(\lambda_ 1\) and sufficient conditions of \(\psi(M)\) for being a totally geodesic sphere are given for general codimension in terms of an eigenfunction belonging to \(\lambda_ 1\).
scientific article; zbMATH DE number 1886789
scientific article