A lower bound for the first eigenvalue in the Laplacian operator on compact Riemannian manifolds

This paper gives a simple proof of the main result of Ling [J. Ling, Lower bounds of the eigenvalues of compact manifolds with positive Ricci curvature, Ann. Global Anal. Geom. 31 (2007) 385–408] in an in-depth study of the sharp lower bound for the first eigenvalue in the Laplacian operator on compact Riemannian manifolds with nonnegative Ricci curvature. Although we use Ling’s methods on the whole, to some extent we deal with the singularity of test functions and greatly simplify many of the calculations involved. This may provide a new way for estimating eigenvalues.