arXiv2026
Several metric splitting theorems under spectral Ricci curvature lower bounds have been established in recent years, yet the existing literature mostly yields only isometric direct product decompositions. In some parameter regimes, direct product splitting is even known to fail. In this paper, we study warped (weighted) area-minimizing hypersurfaces and use their variational properties to prove local warped product metric splitting under spectral Ricci curvature conditions. Given a smooth compact connected $n$-dimensional Riemannian manifold $(M^n,g)$ with non vanishing homology $H_{n-1}(M;\mathbb Z)$, satisfying our spectral Ricci lower bound, and with dimension $3\le n\le 5$, we consider a nontrivial homological warped area-minimizing hypersurface $Σ$ whose existence follows from classical geometric measure theory. By analyzing the first and second variations of the warped area functional together with a foliation argument, we show that $M$ locally decomposes as a warped product $(-ε,ε)\times_f Σ$, where the warping function is given by $f=u^{-1}$ for the positive weight function $u$ appearing in the spectral Ricci condition. Our work complements the existing splitting theory for spectral Ricci curvature: even in parameter ranges where product type splitting cannot hold, we still obtain local warped product rigidity via weighted minimal hypersurfaces.