Scalar Curvature Compactness for Warped Products on ${\mathbb{S}}^2\times {\mathbb{S}}^1$ with Varying Base Metrics

We study the Gromov–Sormani MinA scalar curvature compactness conjecture for warped product metrics on ${\mathbb{S}}^2\times {\mathbb{S}}^1$ of the form introduced by Kazaras-Xu in \cite{KazarasXu2023} as follows: $$g_i={\phi }_i^{-2}{h}_i+{\phi }_i^{2}d{\xi }^2,h_i=d{r}^2+u_i^2(r)d{\theta }^2\mathrm{.}$$ Assuming nonnegative scalar curvature, a uniform volume upper bound, and a positive lower bound for the areas of closed minimal surfaces, we prove a uniform diameter bound for the base surfaces $({\mathbb{S}}^2,h_i)$. Based on this key estimate, we further obtain compactness of the base warping functions $u_i$ and local and global estimates for the fiber warping functions ${\phi }_i$. After passing to a subsequence, the metrics converge in $L^p$, for every finite $p$, to a limit metric $g_{\mathrm{\infty }}$. %on the regular region. We also obtain Gromov–Hausdorff and Sormani–Wenger intrinsic flat subconvergence, and prove that $g_{\mathrm{\infty }}$ has nonnegative scalar curvature in the distributional sense of Lee–LeFloch. Thus the Gromov–Sormani scalar curvature compactness conjecture is verified for this warped product class. Finally, we construct a $C^{1,\alpha }$ example illustrating the subtlety of volume-limit tests for nonnegative scalar curvature in low regularity.