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 $${\lambda }_1(-\gamma \mathrm{\Delta }+\mathrm{R}\mathrm{i}\mathrm{c})\ge 0,H_{\mathrm{\partial }M}\ge 0,$$ and in the sharp range $0\le \gamma <4$ if $n=2$, and $0\le \gamma <\frac{n-1}{n-2}$ if $n\ge 3$, a (possibly noncompact) complete manifold with disconnected boundary, with at least one compact boundary component, must split isometrically as a product $[0,L]\times \mathrm{\Sigma }$. Our second main contribution is a topological rigidity result for the relative fundamental group ${\pi }_1(M,\mathrm{\partial }M)$, combined with a deep theorem of Lawson–Michelsohn. We prove that, in dimensions $n\mathrm{\ne }4$, 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 $\gamma \ge 0$ if $n=2$, and $0\le \gamma \le \frac{n-1}{n-2}$ if $n\ge 3$. This range of $\gamma$ is sharp for the latter result to hold.