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 twosphere. 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 affine monodromy can be shown to lie in SL(3, ℤ)∝ℝ³.) 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 ℂℙ¹ minus three points, and we prove existence of a solution using the direct method in the calculus of variations.