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: $$g_i=\varphi_i^{-2}h_i+\varphi_i^2d\xi^2, \qquad h_i=dr^2+u_i^2(r)d\theta^2.$$ 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 $\varphi_i$. After passing to a subsequence, the metrics converge in $L^p$, for every finite $p$, to a limit metric $g_\infty$. %on the regular region. We also obtain Gromov--Hausdorff and Sormani--Wenger intrinsic flat subconvergence, and prove that $g_\infty$ 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.