Rigidity Results Involving Stabilized Scalar Curvature

We establish a rigidity theorem for Brendle and Hung's recent systolic inequality, which involves Gromov's notion of $T^{\rtimes}$-stabilized scalar curvature. Our primary technique is the construction of foliations by free boundary weighted constant mean curvature hypersurfaces, enabling us to generalize several classical scalar curvature rigidity results to the $T^{\rtimes}$-stabilized setting. Additionally, we develop a monotone quantity using Ricci flow coupled with a heat equation, which is essential for rigidity analysis.