Geometric inequalities and rigidity theorems on equatorial spheres

We prove rigidity for hypersurfaces in the unit (n+1)-sphere whose scalar curvature is bounded below by n(n-1), without imposing constant scalar curvature nor constant mean curvature. The lower bound n(n-1) is critical in the sense that some natural differential operators associated to the scalar curvature may be fully degenerate at geodesic points and cease to be elliptic. We overcome the difficulty by developing an approach to investigate the geometry of level sets of a height function, via new geometric inequalities.