First eigenvalue of submanifolds in Euclidean space
type: Item
instance of: scholarly article
title: First eigenvalue of submanifolds in Euclidean space
zbMATH Open document ID: 0966.53041
DOI: 10.1155/S0161171200003306
author: Kairen Cai
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2001-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/48845
review text: 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)].
reviewed by: Liviu Ornea
Mathematics Subject Classification ID: 53C40
zbMATH DE Number: 1517264
zbMATH Keywords: eigenvalue
zbMATH Keywords: Laplacian
zbMATH Keywords: Ricci curvature
zbMATH Keywords: second fundamental form
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1517264