Erratum to Affine Manifolds, Syz Geometry and the “y” Vertex

We prove the existence of a solution to the Monge-Ampère equation detHess(φ) = 1 on a cone over a thrice-punctured two-sphere. The total space of the tangent bundle is thereby a Calabi-Yau manifold with flat special Lagrangian fibers. (Each fiber can be quotiented to three-torus if the the affine monodromy can be shown to lie in SL(3,Z)⋉ R .) Our method is through Baues and Cortés’s result that a metric cone over an elliptic affine sphere has a parabolic affine sphere structure (i.e., has a Monge-Ampère solution). The elliptic affine sphere structure is determined by a semilinear PDE on CP minus three points, and we prove existence of a solution using the direct method in the calculus of variations.