I have two questions:

1- what is the relation between eigenvalues of geometric operators such as Laplace operator and topology or geometry of a Riemannian manifold?(please give an example if possible).

2- In many papers we see that the author proves the monotonicity of some quantities where evolve along a geometric flow such as Ricci flow. For example Cao stated the following in his paper: where $r$ is the average scalar curvature in the above Corollary. I want to know what is the application of the Corollary 2.4 in the study of the topology or geometry of the manifold $M$?