Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds.
type: Item
instance of: scholarly article
title: Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds.
zbMATH Open document ID: 1050.58024
author: Jean-François Grosjean
published in: Pacific Journal of Mathematics
publication date: 2003-02-13T00:00:00Z
review text: Let \((M,g)\) be an \(m\) dimensional compact Riemannian manifold which is isometrically immersed in a simply connected space form. Let \(\lambda_1\) be the first non-zero eigenvalue of the scalar Laplacian. The author gives optimal upper bounds for \(\lambda_1\) in terms of \(r\)-th mean curvatures of the embedding and the scalar curvature. This generalizes previous work of \textit{R. C. Reilly} [Comment. Math. Helv. 52, 525--533 (1977; Zbl 0382.53038)] in flat space to the other space forms -- i.e. 2 point homogeneous spaces. The author shows that if \(M\) is a compact hyper surface of positive scalar curvature immersed in Euclidean space \(\mathbb{R}^{m+1}\) and if \(g\) is a Yamabe metric, then \(M\) is a standard sphere.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1868175
zbMATH Keywords: space form
zbMATH Keywords: mean curvature
MaRDI profile type: Publication
full work available at URL: pjm.2002.206.93
OpenAlex ID: W2040922692
DOI: 10.2140/PJM.2002.206.93
description: scientific article; zbMATH DE number 1868175