### Abstract

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)∝ℝ^{3}.) 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^{1} minus three points, and we prove existence of a solution using the direct method in the calculus of variations.

Original language | English (US) |
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Pages (from-to) | 129-158 |

Number of pages | 30 |

Journal | Journal of Differential Geometry |

Volume | 71 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Journal of Differential Geometry*,

*71*(1), 129-158. https://doi.org/10.4310/jdg/1143644314