On the Stability of Llarull's Theorem in Dimension Three

Brian Allen; Edward Bryden; Demetre Kazaras

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arXiv · 2305.18567

On the Stability of Llarull's Theorem in Dimension Three

Brian Allen·Edward Bryden·Demetre Kazaras

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Abstract

Llarull's Theorem states that any Riemannian metric on the $n$-sphere which has scalar curv{\-}ature 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 {\emph{Spherical Stability Problem}}, which probes the flexibility of this fact. We give a resolution to this problem in dimension $3$. Informally, the main result asserts 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 using spacetime harmonic functions.

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Brian Allen, Edward Bryden, Demetre Kazaras. 2023-05-29. On the Stability of Llarull's Theorem in Dimension Three. https://arxiv.org/abs/2305.18567

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