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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: Corollry 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 $M$?

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  • $\begingroup$ Isospectrality is in general weaker than isometry. Nonetheless, one can often recover "geometric aggregates" (such as volume or total curvature) and topological quantities (such as Reidemeister torsion) from the asymptotic behavior of eigenvalues of the Laplacian. The lowest positive eigenvalue is also a quantity of interest and can be related to the geometry of the manifold by a theorem of Cheeger. $\endgroup$ – Neal Jan 8 at 19:44
  • $\begingroup$ This question is probably too broad. The spectral theory of differential operators is a massive subject, even if one restricts to questions regarding the Laplacian on closed manifolds. You might explore the book of Zelditch sites.math.northwestern.edu/~zelditch/Eigenfunction.pdf to refine your question. $\endgroup$ – Andy Sanders Jan 8 at 19:45
  • $\begingroup$ Would you please give a reference for the Cheeger theorem? $\endgroup$ – user162551 Jan 8 at 20:00
  • $\begingroup$ @user162551 mathscinet.ams.org/mathscinet-getitem?mr=0402831 and an inequality in the other direction by Buser mathscinet.ams.org/mathscinet-getitem?mr=0683635 $\endgroup$ – Neal Jan 8 at 21:03

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