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 $X \times \mathbb{ R}^3$, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold $X \times \mathbb{ C}$, where $X$ is an $A_2$-type ALE hyperk\"ahler 4-manifold. We prove this conjecture by solving a real Monge-Amp\`ere 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 $X\times \mathbb{ C}$, where $X$ arises from the Gibbons-Hawking construction.
Categories: math.DG math.AP math.SG