On a Rigidity Result in Positive Scalar Curvature Geometry

TL;DR

This paper establishes a rigidity theorem for scalar curvature on spheres, showing that certain geodesic balls are uniquely determined by their scalar curvature and boundary conditions, refining previous conjectures in the field.

Contribution

The paper proves a new scalar curvature rigidity theorem for geodesic balls in spheres, using real Killing connections and Dirac operators, refining the Min-Oo conjecture.

Findings

Rigidity holds for geodesic balls of radius less than π/2 in spheres.

Rigidity fails for the hemisphere.

The proof involves real Killing connections and boundary value problems.

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 $n+1~(n\geq 2)$ 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.