On the Donaldson–Scaduto conjecture

Habibi Esfahani, Saman; Li, Yang

doi:10.2140/gt.2026.30.959

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On the Donaldson–Scaduto conjecture

Saman Habibi Esfahani and Yang Li

Geometry & Topology 30 (2026) 959–981

DOI: 10.2140/gt.2026.30.959

Abstract

Motivated by $G_{2}$ -manifolds with coassociative fibrations in the adiabatic limit, Donaldson and Scaduto conjectured the existence of associative submanifolds homeomorphic to a three-holed $3$ -sphere with three asymptotically cylindrical ends in the $G_{2}$ -manifold $X \times ℝ^{3}$ , or equivalently similar special Lagrangians in the Calabi–Yau 3-fold $X \times ℂ$ , where $X$ is an $A_{2}$ -type ALE hyperkähler 4-manifold. We prove this conjecture by solving a real Monge–Ampère equation with a singular right-hand side, which produces a potentially singular special Lagrangian. Then, we prove the smoothness and asymptotic properties for the special Lagrangian using inputs from geometric measure theory. The method produces many other asymptotically cylindrical $U (1)$ -invariant special Lagrangians in $X \times ℂ$ , where $X$ arises from the Gibbons–Hawking construction.

Keywords

$G_2$-manifolds, coassociative fibrations, associative submanifolds, special Lagrangians, Calabi–Yau 3-folds

Mathematical Subject Classification

Primary: 32Q25, 53C26, 53C29, 53C38, 53D20

References

Publication

Received: 11 February 2024

Revised: 14 April 2025

Accepted: 20 July 2025

Published: 20 April 2026

Proposed: András I Stipsicz

Seconded: Simon Donaldson, Ciprian Manolescu

Authors

Saman Habibi Esfahani
Department of Mathematics Duke University Durham, NC United States	Center of Mathematical Sciences and Applications Harvard University Cambridge, MA United States

Yang Li
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States	Department of Pure Mathematics and Mathematical Statistics Cambridge University Cambridge United Kingdom

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Volume 30, issue 3 (2026)