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Donaldson’s program on the adiabatic limit of K3-fibered G₂-manifolds relates associative submanifolds to certain weighted trivalent graphs on the base called gradient cycles. A key ingredient in this relation, near a trivalent vertex, is the existence and uniqueness of an associative pair of pants in the G₂-manifold formed as a product of a K3 surface and the Euclidean 3-plane. This is known as the Donaldson–Scaduto conjecture. Although this conjecture remains open, a local version replacing the K3 surface with an ALE or ALF hyperkähler 4-manifold of type A₂, has been shown to exist by Esfahani and Li. In this talk, I will discuss our joint work on proving the uniqueness, showing that no other associative pair of pants satisfies this local version of the conjecture. |