On the Donaldson-Scaduto conjecture

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 $X \times \mathbb{R}^3$, or equivalently similar special Lagrangians in the Calabi-Yau 3-fold $X \times \mathbb{C}$, where $X$ is an $A_2$-type ALE 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 $X\times \mathbb{C}$, where $X$ arises from the Gibbons-Hawking construction.

Figures (12)

• Figure 1: Donaldson-Scaduto conjecture.
• Figure 2: $L_{i,\text{red}} \subset \mathbb{R}^4 = \mathbb{R}^2_{(u_1,u_2)} \times \mathbb{R}^2_{(y_1,y_2)}$.
• Figure 3: $\pi_1( \cup L_{i,\text{red}})$ (left) and $\pi_2(\cup L_{i,\text{red}} )$ (right).
• Figure 4: $L_{\text{red}}$ in $Z_{\text{red}}= \mathbb{R}^4$, in the case $n=3$.
• Figure 5: Approximating domains $U_t \to U$, in the case $n=3$.

Theorems & Definitions (38)

• Conjecture 1: Donaldson-Scaduto
• Theorem 1: Donaldson-Scaduto conjecture, local version
• Theorem 2: Generalization
• Remark 1
• Remark 2
• Theorem 3: Dirichlet problem
• proof
• Lemma 1: Rauch-Taylor MR0454331
• Lemma 2
• proof