Journal of Differential Geometry

Affine manifolds, SYZ geometry and the "Y" vertex

John Loftin, Shing-Tung Yau, and Eric Zaslow

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We prove the existence of a solution to the Monge-Ampère equation detHess(ø) = 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 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.

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J. Differential Geom., Volume 71, Number 1 (2005), 129-158.

First available in Project Euclid: 29 March 2006

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Loftin, John; Yau, Shing-Tung; Zaslow, Eric. Affine manifolds, SYZ geometry and the "Y" vertex. J. Differential Geom. 71 (2005), no. 1, 129--158. doi:10.4310/jdg/1143644314.

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