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 $λ_1(M)$ of the Laplace--Beltrami operator, subject to the Neumann boundary condition when $\partial M\neq\emptyset$. Given a reference scalar cur