On the Donaldson-Scaduto conjecture

1. Problem Statement and Context

The paper addresses a conjecture proposed by Simon Donaldson and Scaduto regarding the existence of specific associative submanifolds in $G_2$-manifolds and special Lagrangian submanifolds in Calabi-Yau 3-folds.

• Global Context: In the study of $G_2$-manifolds constructed via coassociative K3 fibrations in the adiabatic limit, Donaldson and Scaduto conjectured that associative submanifolds can form "triple junctions" where three cylindrical ends meet.
• Local Model: The paper focuses on the local version of this conjecture. The ambient space is replaced by $X \times \mathbb{R}^3$, where $X$ is an $A_2$-type ALE (Asymptotically Locally Euclidean) gravitational instanton (a hyperkähler 4-manifold).
• The Conjecture: There exists an associative submanifold $P \subset X \times \mathbb{R}^3$ that is homeomorphic to a three-holed 3-sphere ($S^3$ with three punctures) and has three ends asymptotic to specific half-cylinders $\Sigma_i \times (\mathbb{R}^+ v_i)$.
• Equivalence: By identifying $X \times \mathbb{R}^3$ with $X \times \mathbb{R}^2 \times \mathbb{R}$, the problem is reformulated in terms of special Lagrangian submanifolds in the Calabi-Yau 3-fold $Z = X \times \mathbb{R}^2$. The authors generalize the problem to $A_{n-1}$-type instantons, seeking an $(n-1)$-dimensional family of special Lagrangians homeomorphic to an $n$-holed 3-sphere with $n$ asymptotically cylindrical ends.

2. Methodology

The authors employ a combination of geometric analysis, partial differential equations (PDEs), and geometric measure theory. The strategy proceeds in four main stages:

A. Dimensional Reduction

Assuming $U(1)$-invariance (motivated by the Gibbons-Hawking ansatz for $X$), the problem is reduced from a 6-dimensional special Lagrangian condition to a 2-dimensional problem on the symplectic quotient $Z_{red} \cong \mathbb{R}^4$.

• The special Lagrangian condition reduces to a system of equations for a map $F: U \subset \mathbb{R}^2 \to \mathbb{R}^2$, where $U$ is the convex polygon with vertices $p_1, \dots, p_n$ (the monopole points).
• Under a graphical assumption ($F = \nabla \phi$), the system reduces to a real Monge-Ampère equation:
  $$\det(D^2 \phi) = V(u) = A + \sum_{i=1}^n \frac{1}{2|u - p_i|}$$
  where $V$ is the harmonic function defining the Gibbons-Hawking metric.

B. Solving the Dirichlet Problem

The authors solve the Monge-Ampère equation on the convex polygon $U$ with specific boundary conditions:

• Boundary Data: On each edge $[p_i, p_{i+1}]$, the potential $\phi$ is required to be affine linear, corresponding to the asymptotic positions of the cylindrical ends.
• Singularities: The right-hand side $V$ has singularities at the vertices $p_i$.
• Technique: They use an approximation strategy, solving the equation on strictly convex domains $U_t \subset U$ and taking the limit $t \to 1$. They establish uniform $C^0$ bounds and Lipschitz continuity to ensure the existence of a continuous convex solution $\phi$ that is smooth in the interior.

C. Analysis of Gradient Behavior

A critical step is understanding the behavior of the gradient $\nabla \phi$ near the boundary:

• Near Edges: The normal derivative $\nabla_{normal} \phi$ diverges to $+\infty$ as one approaches the open edges, while the tangential derivative converges to a constant. This ensures that the edges of the polygon do not contribute to the closure of the submanifold.
• Near Vertices: The gradient map $\nabla \phi$ maps the interior of $U$ to $\mathbb{R}^2 \setminus \bigcup C_{p_i}$, where $C_{p_i}$ are disjoint convex sets (subgradients) associated with the vertices. The authors prove that the image of the gradient avoids these sets, which is crucial for the topology of the resulting submanifold.

D. Regularity and Topology via Geometric Measure Theory

To prove the constructed object is a smooth submanifold:

• Closure: They take the closure of the open special Lagrangian $L^\circ$.
• Singularities: Using Allard's Regularity Theorem and Joyce's classification of $U(1)$-invariant special Lagrangian cones in $\mathbb{C}^3$, they prove that the tangent cones at the points lying above the vertices $p_i$ are smooth 3-planes with multiplicity one. This rules out singularities, proving the closure $L$ is a smooth submanifold.
• Asymptotics: They establish exponential decay estimates, showing $L$ is a small perturbation of the model cylinders at infinity.
• Topology: Using the Van Kampen theorem and properties of the fibration, they prove that the resulting manifold is simply connected and thus homeomorphic to an $n$-holed 3-sphere.

3. Key Contributions and Results

• Proof of the Local Donaldson-Scaduto Conjecture: The paper rigorously proves the existence of the $A_2$-type local model for the triple junction of associative submanifolds.
• Generalization to $A_{n-1}$: The result is extended to $A_{n-1}$-type ALE/ALF spaces, constructing an $(n-1)$-dimensional family of special Lagrangians homeomorphic to an $n$-holed 3-sphere.
• Parameter Count: The construction depends on $n-1$ real parameters (subject to the constraint $\sum c_i = 0$), which correspond to the translations of the asymptotic ends. The authors note these deformations are rigid once asymptotic conditions are fixed.
• Analytical Breakthrough: The paper provides a robust method for solving the real Monge-Ampère equation with singular right-hand sides and affine boundary data on convex polygons, specifically handling the gradient divergence near edges and the subgradient structure at vertices.
• Smoothness via Classification: The proof of smoothness relies on a deep application of Joyce's classification of $U(1)$-invariant cones to rule out non-smooth tangent cones at the "corners" of the domain.

4. Significance

• Building Blocks for Gluing: These special Lagrangians serve as fundamental building blocks for constructing new special Lagrangians in "local Calabi-Yau 3-folds" via gluing techniques. This is essential for the broader program of constructing compact $G_2$-manifolds and understanding their moduli spaces.
• Adiabatic Limit Understanding: The work provides a concrete mathematical realization of the heuristic picture proposed by Donaldson and Scaduto regarding how associative submanifolds behave in the adiabatic limit of $G_2$-manifolds (specifically near triple junctions).
• Interplay of Fields: The paper demonstrates a powerful synthesis of hyperkähler geometry, Calabi-Yau structures, nonlinear PDEs (Monge-Ampère), and geometric measure theory (regularity of currents).

In summary, Habibi Esfahani and Li provide a complete existence proof for a specific class of singular-looking but actually smooth special Lagrangians, resolving a key conjecture in the study of $G_2$-geometry and providing new tools for the construction of calibrated submanifolds.