First eigenvalue of submanifolds in Euclidean space
Let \(M^n\) be a compact submanifold immersed in the Euclidean space \(\mathbb R^{n+p}\). Denote by \(\lambda_1\) the first eigenvalue of the Laplacian on \(M\) and by \(\sigma\) the second fundamental form. The main result of this note is that \(\lambda_1\leq \text{max}_{M}|\sigma|^2\). Moreover, if \(\lambda_1\geq |\sigma|^2\) in one point, then \(M\) is isometric to the sphere \(S^n(1)\). As a consequence, a closed, convex hypersurface of \(\mathbb R^{n+1}\) with scalar curvature \(R\leq (n-1)\lambda_1\) is isometric to a \(S^n(r)\). This generalizes and improves a result by \textit{S. Deshmukh}, [Q. J. Math. Oxf., II. Ser. 49, 35-41 (1998; Zbl 0906.53003)]. Another application of the main result leads to a nonimmersion theorem of Chern-Kuiper type, relaxing the assumption of a similar statement in \textit{S. Deshmukh} and \textit{M. A. Al-Gwaiz} [Q. J. Math. Oxf., II Ser. 45, 151-157 (1994; Zbl 0810.53046)].
scientific article; zbMATH DE number 1517264