### 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

}

*Journal of Differential Geometry*, vol. 71, no. 1, pp. 129-158. https://doi.org/10.4310/jdg/1143644314

**Affine manifolds, SYZ geometry and the “Y” vertex.** / Loftin, John; Yau, Shing Tung; Zaslow, Eric.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Affine manifolds, SYZ geometry and the “Y” vertex

AU - Loftin, John

AU - Yau, Shing Tung

AU - Zaslow, Eric

PY - 2005/1/1

Y1 - 2005/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=61849112720&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=61849112720&partnerID=8YFLogxK

U2 - 10.4310/jdg/1143644314

DO - 10.4310/jdg/1143644314

M3 - Article

VL - 71

SP - 129

EP - 158

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -