Spectral Gap Estimates on Nearly Einstein Manifolds with Integral Curvature Deviation

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 $\mathrm{\partial }M\mathrm{\ne }\mathrm{\varnothing }$. Given a reference scalar curvature $R_E>0$, set $K_E:=R_E\mathrm{/}(n(n-1))$, and define the traceless Ricci tensor $\mathring{\text{Ric}}:=\text{Ric}-\frac{R}{n}g$ and the Einstein deviation $$\mathrm{\Theta }:=\mathrm{\mid }\mathring{\text{Ric}}\mathrm{\mid }+\frac{\mathrm{\mid }R-R_E\mathrm{\mid }}{n},{\kappa }_{\mathrm{E}\mathrm{i}\mathrm{n}}(p):={\left(\frac{1}{\text{vol}(M)}{\int }_M{\mathrm{\Theta }}^{p}dV\right)}^{1\mathrm{/}p},p>n\mathrm{/}2.$$ We say that $M$ is $\epsilon$-nearly Einstein if ${\kappa }_{\mathrm{E}\mathrm{i}\mathrm{n}}(p)<\epsilon$. 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 $\text{diam}(M)\le D$ and ${\kappa }_{\mathrm{E}\mathrm{i}\mathrm{n}}(p)<{\epsilon }_0$, then $${\lambda }_1(M)\ge \alpha {\lambda }_1(n,K_E,D),$$ 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 $D\ge \pi \mathrm{/}\sqrt{K}$. The proof combines the Bochner formula for the decomposition $\text{Ric}=(n-1)K_{E}g+\theta g+\mathring{\text{Ric}}$ with an auxiliary function $J$ satisfying $$\mathrm{\Delta }J-\tau \frac{\mathrm{\mid }\mathrm{\nabla }J{\mathrm{\mid }}^2}{J}-2J\mathrm{\Theta }=-\sigma J$$ and Neumann boundary conditions, together with Moser iteration and a maximum principle. As ${\kappa }_{\mathrm{E}\mathrm{i}\mathrm{n}}(p)\to 0$, the estimate recovers the classical Bakry–Qian theorem.