Open Access
September 2005 Affine manifolds, SYZ geometry and the "Y" vertex
John Loftin, Shing-Tung Yau, Eric Zaslow
J. Differential Geom. 71(1): 129-158 (September 2005). DOI: 10.4310/jdg/1143644314


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.


Download Citation

John Loftin. Shing-Tung Yau. Eric Zaslow. "Affine manifolds, SYZ geometry and the "Y" vertex." J. Differential Geom. 71 (1) 129 - 158, September 2005.


Published: September 2005
First available in Project Euclid: 29 March 2006

zbMATH: 1094.32007
MathSciNet: MR2191770
Digital Object Identifier: 10.4310/jdg/1143644314

Rights: Copyright © 2005 Lehigh University

Vol.71 • No. 1 • September 2005
Back to Top