Geometric inequalities and rigidity theorems on equatorial spheres

TL;DR

This paper establishes rigidity results for hypersurfaces in spheres and Euclidean space with scalar curvature bounds, using new geometric inequalities to handle degenerate points and boundary conditions.

Contribution

It introduces novel geometric inequalities to prove rigidity of hypersurfaces with boundary in spheres, especially at critical scalar curvature levels.

Findings

Hypersurfaces with boundary in spheres are part of equatorial spheres under scalar curvature bounds.

New inequalities help analyze level sets of height functions at degenerate points.

Rigidity results for hyperplanes and cylinders in Euclidean space are also derived.

Abstract

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.