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Geometric inequalities and rigidity theorems on equatorial spheres

We prove rigidity for hypersurfaces with boundary in the unit $(n+1)$-sphere with scalar curvature bounded below by $n(n-1)$. Under appropriate boundary conditions, the hypersurfaces are shown to be part of the equatorial spheres. The lower bound $n(n-1)$ is critical in the sense that the hypersurface may contain geodesic points and some natural differential operators are fully degenerate at geodesic points. We overcome the difficulty by studying the geometry of level sets of a height function, via new geometric inequalities. Some rigidity results of hyperplanes and generalized cylinders are also obtained for hypersurfaces with boundary and with nonnegative scalar curvature in Euclidean space.