Affine manifolds, SYZ geometry and the “Y” vertex

John Loftin, Shing Tung Yau, Eric Zaslow

Research output: Contribution to journalArticle

17 Citations (Scopus)

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)∝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.

Original languageEnglish (US)
Pages (from-to)129-158
Number of pages30
JournalJournal of Differential Geometry
Volume71
Issue number1
DOIs
StatePublished - Jan 1 2005

Fingerprint

Vertex of a graph
Cone
Fiber
Calabi-Yau Manifolds
Tangent Bundle
Monodromy
Calculus of variations
Direct Method
Semilinear
Torus
Metric

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology

Cite this

Loftin, John ; Yau, Shing Tung ; Zaslow, Eric. / Affine manifolds, SYZ geometry and the “Y” vertex. In: Journal of Differential Geometry. 2005 ; Vol. 71, No. 1. pp. 129-158.
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Affine manifolds, SYZ geometry and the “Y” vertex. / Loftin, John; Yau, Shing Tung; Zaslow, Eric.

In: Journal of Differential Geometry, Vol. 71, No. 1, 01.01.2005, p. 129-158.

Research output: Contribution to journalArticle

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