Affine manifolds, SYZ geometry and the "Y" vertex
Loftin, John ; Yau, Shing-Tung ; Zaslow, Eric
J. Differential Geom., Tome 69 (2005) no. 3, p. 129-158 / Harvested from Project Euclid
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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.
Publié le : 2005-09-14
Classification:  
@article{1143644314,
     author = {Loftin, John and Yau, Shing-Tung and Zaslow, Eric},
     title = {Affine manifolds, SYZ geometry and the "Y" vertex},
     journal = {J. Differential Geom.},
     volume = {69},
     number = {3},
     year = {2005},
     pages = { 129-158},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1143644314}
}

Loftin, John; Yau, Shing-Tung; Zaslow, Eric. Affine manifolds, SYZ geometry and the "Y" vertex. J. Differential Geom., Tome 69 (2005) no. 3, pp.  129-158. http://gdmltest.u-ga.fr/item/1143644314/