2024/01/27 by Esfahani, Saman Habibi, Li, Yang
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.2401.15432
Motivated by G2-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 G2-manifold X × ℝ3, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold X × ℂ, where X is an A2-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× ℂ, where X arises from the Gibbons-Hawking construction.