Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation

Abstract

Let $(M^n,g)$ be a compact Riemannian manifold of dimension $n\ge 2$, possibly with nonempty convex $C^2$ boundary. We study the first nonzero eigenvalue ${\lambda }_1(M)$ of the Laplace--Beltrami operator, subject to the Neumann boundary condition when $\partial M\ne \varnothing$. Given a reference scalar curvature $R_E>0$, set $K_E:=R_E/(n(n-1))$, and define the traceless Ricci tensor $\mathring{\Ric}:=\Ric-\frac{R}{n}g$ and the Einstein deviation \[ Θ:=|\mathring{\Ric}|+\frac{|R-R_E|}{n},\qquad κ_{\mathrm{Ein}}(p):=\left(\frac{1}{\vol(M)}\int_MΘ^p\,dV\right)^{1/p}, \quad p>n/2. \] We say that $M$ is $\epsilon$-nearly Einstein if ${\kappa }_{\mathrm{Ein}}(p)<\epsilon$. The main result states that for every $\alpha \in (0,1)$ there exists ${\epsilon }_0={\epsilon }_0(n,p,\alpha ,D,R_E)>0$ such that if $\diam(M)\le D$ and ${\kappa }_{\mathrm{Ein}}(p)<{\epsilon }_0$, then \[ λ_1(M)\ge α\,λ_1(n,K_E,D), \] where ${\lambda }_1(n,K,D)$ is the first nonzero eigenvalue of the Bakry--Qian one-dimensional model problem with curvature parameter $K$, with the convention ${\lambda }_1(n,K,D)=nK$ when $K>0$ and $D\ge \pi /\sqrt{K}$. The proof combines the Bochner formula for the decomposition $\Ric=(n-1)K_Eg+θg+\mathring{\Ric}$ with an auxiliary function $J$ satisfying \[ ΔJ-τ\frac{|\nabla J|^2}{J}-2JΘ=-σJ \] and Neumann boundary conditions, together with Moser iteration and a maximum principle. As ${\kappa }_{\mathrm{Ein}}(p)\to 0$, the estimate recovers the classical Bakry--Qian theorem.