On a Rigidity Result in Positive Scalar Curvature Geometry

Abstract

I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\frac{\pi }{2}$ in $n+1(n\ge 2)$ dimensional unit sphere can be rigid under smooth deformations that increase scalar curvature preserving the intrinsic geometry and the mean curvature of the boundary, and such rigidity result fails for the hemisphere. The proof of this assertion requires the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. The result serves as the sharpest refinement of the now-disproven Min-Oo conjecture.

arXiv commentThe result is far from optimal; the radius of the geodesic ball that can be rigid under large deformation is not clear at all, making the result almost useless despite some new techniques such as using the biorthogonal spin system. Therefore, it is best to retract this