Sormani's minA-IF stability conjecture for Riemannian 3-tori

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Let {(Mi,gi)}i=1∞\{(M_i,g_i)\}_{i=1}^\infty be a sequence of Riemannian 33-tori. Assume that diam⁡(Mi,gi)\operatorname{diam}(M_i,g_i) is uniformly bounded above, while vol⁡(Mi,gi)\operatorname{vol}(M_i,g_i) and minA⁡(Mi,gi)\operatorname{minA}(M_i,g_i) are uniformly bounded away from 00. Here, minA⁡(Mi,gi)\operatorname{minA}(M_i,g_i) is the infimum of the areas of closed embedded minimal surfaces in MiM_i. Sormani's minA-IF stability conjecture. If Rgi⩾−1/iR_{g_i}\geqslant-1/i, then (Mi,gi)(M_i,g_i) has a subsequence converging to a flat torus in the volume-preserving intrinsic flat sense as i→∞i\to\infty. This conjecture seeks an almost-rigidity statement for the Geroch conjecture that excludes thin tunnels and other-world constructions through a uniform lower bound on the minA invariant. The source does not state whether the conjecture has been resolved.

References

Primary source

Demetre Kazaras and Kai Xu, “Drawstrings and flexibility in the Geroch conjecture”, arXiv:2309.03756 (2026).

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