Monotonicity Properties of the First Eigenvalue of the Laplacian Operator on Ricci Solitons

Abstract

In this paper, we deal with the monotonicity properties of the first eigenvalue of the Laplacian operator on Ricci solitons. Firstly, by using the monotonicity formula of the $\mathcal{F}$ functional, we derive a monotonicity formula of the first eigenvalue of the Laplacian operator on Ricci solitons. Based on this, we also prove an exponential decreasing property of the first eigenvalue.