Warped area-minimizing hypersurface and warped product metric

Let $(M^n,g)$ be a smooth, compact, connected and oriented Riemannian manifold with $3\le n\le 7$ and $H_{n-1}(M;\mathbb{Z})\neq 0$, and let $u>0$ be smooth on $M$. Suppose that \[ -γu^{-1}Δ_g u+\Ricmin_g\ \geq\ (n-1)(n-3)\,u^{-2}|\nabla u|^2, \qquad γ=n-1, \] where $\Ricmin_g$ is the pointwise least Ricci curvature. Since both this hypothesis and the class of warped (weighted) area-minimizers are invariant under $u\mapstoλu$, we normalize $u\equiv1$ on the minimizer. We prove that some non-trivial class in $H_{n-1}(M;\mathbb{Z})$ contains a warped area-minimizing hypersurface $Σ$ on which $u$ is constant, and that $M$ locally splits as a warped product $(-ε,ε)\times_fΣ$, $g=dt^2+f^2(t)g_Σ$, with $f=u^{-1}$ and $\operatorname{Ric}_{g_Σ}\geq (n-2)[(f')^2-ff'']\,g_Σ$. By extending the foliation via Federer--Fleming compactness of the constant-warped-area leaves, this local splitting upgrades to a \emph{global} one: $M$ is covered by $\mathbb{R}\times_fΣ$, its universal cover is $\mathbb{R}\times_f\tildeΣ$ where $\tildeΣ$ is the universal cover of $Σ$, and $f=u^{-1}$ is periodic and bounded above and below, so $M$ fibres over $\mathbb{S}^1$ with fibre a quotient of $Σ$. This spectral bound is saturated with equality by warped product metrics, which is what makes it the natural threshold for warped rather than direct product rigidity. When $u\equiv1$ it reduces to $\Ricmin_g\geq0$ and our conclusion to the classical product splitting.