Upper bounds for the first eigenvalue of the Laplacian on compact submanifolds. (Q1858312)

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.