Applications of Bochner formula to minimal submanifold of the sphere.
type: Item
instance of: scholarly article
title: Applications of Bochner formula to minimal submanifold of the sphere.
zbMATH Open document ID: 1033.53047
DOI: 10.1016/S0393-0440(02)00061-X
author: Yanyan Li
published in: Journal of Geometry and Physics
publication date: 2003-03-23T00:00:00Z
review text: 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 \(\lambda_ 1\).
reviewed by: Miyuki Koiso
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 1886789
zbMATH Keywords: minimal submanifolds
zbMATH Keywords: eigenvalues
zbMATH Keywords: second fundamental form
MaRDI profile type: Publication
cites work: Q4522318
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description: scientific article; zbMATH DE number 1886789