Affine manifolds, SYZ geometry and the "Y" vertex

Abstract

We prove the existence of a solution to the Monge-Ampere equation detHess(o) = 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 affine monodromy can be shown to lie in SL(3,Z) × R3.) Our method is through Baues and Cortes's result that a metric cone over an elliptic affine sphere has a parabolic affine sphere structure (i.e., has a Monge-Ampere 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.