Almost Rigidity of the Llarull Theorem

Llarull's Theorem states that any Riemannian metric on the $n$-sphere which has scalar
curvature greater than or equal to $n(n-1)$, and whose distance function is bounded below
by the unit sphere's, is isometric to the unit sphere. Gromov later posed the Spherical
Stability Problem, probing the flexibility of this fact, which we give a resolution of in
dimension $3$. We show that a sequence of Riemannian $3$-spheres whose distance
functions are bounded below by the unit sphere's with uniformly bounded Cheeger
isoperimetric constant and scalar curvatures tending to $6$ must approach the round $3$-
sphere in the volume preserving Sormani-Wenger Intrinsic Flat sense. The argument is
based on a proof of Llarull's Theorem due to Hirsch-Kazaras-Khuri-Zhang
(https://arxiv.org/abs/2209.12857) using spacetime harmonic functions and a
characterization of Sormani-Wenger Intrinsic Flat convergence given by Allen-Perales-
Sormani (https://arxiv.org/abs/2003.01172). This is joint work with E. Bryden and D. Kazaras
(https://arxiv.org/abs/2305.18567).