Lower bound estimates of the first eigenvalue for the \(f\)-Laplacian and their applications (Q2871295)

Let \(( M^n,g)\) be an \(n\)-dimensional closed Riemannian manifold. There are various results on lower bounds for the first eigenvalue of the Laplacian on \(M\) and the authors summarize some of the results relevant to their study. The focus of this paper is to find lower bounds for the first nonzero eigenvalue \(\lambda_1(M^n)\) of the \(f\)-Laplacian \(\Delta_f:=\Delta -\nabla f\cdot \nabla \) with \(N\)-Bakry-Émery Ricci tensor \(\mathrm{Ric}_{f}^{N}:=\mathrm{Ric}+\mathrm{Hess}\, f-\frac{1}{N}\) \(df\otimes df\), defining \(\mathrm{Ric}_{f}=\mathrm{Ric}_f^\infty =\mathrm{Ric}+\mathrm{Hess}\, f\). The authors obtain lower bounds for the first eigenvalue when \(\mathrm{Ric}_f^N\geq K\), \(K\in \mathbb R\). They then obtain corresponding results when \(\mathrm{Ric}_f\geq K\), \(K\in \mathbb R\). They derive as well a lower bound for the diameter of \(M\) when \((M^n,g)\) is a compact nontrivial Ricci soliton, hence a gradient contracting Ricci soliton with \(\mathrm{Ric}_f=Kg\) and \(K>0\).