Rigidity and flexibility under spectral Ricci lower bounds and mean-convex boundary

We study Riemannian manifolds $(M^n,g)$ with mean-convex boundary whose Ricci curvature is nonnegative in a spectral sense. Our first main result is a sharp spectral extension of a rigidity theorem by Kasue: we prove that under the conditions $$\lambda_1(-\gamma\Delta+\mathrm{Ric})\geq 0,\qquad H_{\partial M}\geq 0,$$ and in the sharp range $0\leq \gamma<4$ if $n=2$, and $0\leq\gamma<\frac{n-1}{n-2}$ if $n\geq3$, a (possibly noncompact) complete manifold with disconnected boundary, with at least one compact boundary component, must split isometrically as a product $[0,L]\times \Sigma$. Our second main contribution is a topological rigidity result for the relative fundamental group $\pi_1(M,\partial M)$, combined with a deep theorem of Lawson--Michelsohn. We prove that, in dimensions $n\neq4$, any compact manifold with boundary satisfying the two inequalities above, with at least one of them strict, admits a metric with positive sectional curvature and strictly mean-convex boundary, provided $\gamma\geq0$ if $n=2$, and $0\leq\gamma\leq\frac{n-1}{n-2}$ if $n\geq3$. This range of $\gamma$ is sharp for the latter result to hold.