In this paper, we prove a scalar curvature rigidity result for geodesic balls in $S^n$. This result contrasts sharply with the counterexamples to Min-Oo’s conjecture constructed in "Deformations of the hemisphere that increase scalar curvature," Invent. Math. 185 (2011), 175–197.
"Scalar Curvature Rigidity of Geodesic Balls in $S^n$." J. Differential Geom. 88 (3) 379 - 394, July 2011. https://doi.org/10.4310/jdg/1321366355