Eigenvalues and eigenfunctions of Riemannian manifolds

Abstract

S.-Y. Cheng [Proc. Amer. Soc. 55 (1976), 379-381] investigated closed two-dimensional Riemannian manifolds of genus zero which admit m first eigenfunctions with constant square sum, m > 1 m > 1 . In this note, we will investigate n-dimensional Riemannian manifolds with m eigenfunctions, corresponding to the eigenvalue λ \lambda , and with constant square sum. Examples of such manifolds are minimal submanifolds of spheres. While Cheng investigated closed manifolds, most of our results have local character. We give lower bounds for λ \lambda by curvature functions (scalar curvature, sectional curvature) and apply these results in two cases: (i) to characterize manifolds which are locally isometric to spheres; (ii) to the investigation of minimal submanifolds of spheres. These results extend earlier results of Lange and Simon.