Some Asymptotic Behavior of the First Eigenvalue along the Ricci Flow

The study of behavior of the eigenvalues of differential operators along the flow of metrics is very active. We list a few such studies as follows. Perelman [9] proved the monotonicity of the first eigenvalue of the operator −∆ + 1 4 R along the Ricci flow by using his entropy and was then able to rule out nontrivial steady or expanding breathers on compact manifolds. X. Cao [1] and J. F. Li [6] studied the eigenvalues of −∆+ 1 2 R along the unnormalized Ricci flow and gave some geometric applications. The author [7] studied the eigenvalues of Laplacians of the normalized Ricci flow of metrics and gave a Faber-Krahn type of comparison theorem and a sharp bound of the first nonzero eigenvalue on compact 2-manifolds with negative Euler Characteristic. In [8], the author constructed a class of monotonic quantities along the normalized Ricci flow. In this short note, we study some asymptotic behavior of the first nonzero eigenvalue of the Lalacian along the normalized Ricci flow and give a direct short proof for an asymptotic upper limit estimate. There are more developments to follow this.