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

Abstract

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: \[ g_i=φ_i^{-2}h_i+φ_i^2dξ^2, \qquad h_i=dr^2+u_i^2(r)dθ^2. \] 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 ${\phi }_i$. 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.

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