ar X iv : m at h / 04 05 06 1 v 1 [ m at h . D G ] 4 M ay 2 00 4 AFFINE MANIFOLDS , SYZ GEOMETRY AND THE “ Y ” VERTEX

We study the real Monge-Ampère equation in two and three dimensions, both from the point of view of the SYZ conjecture , where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holo-morphic function representation to Hitchin's description of special Lagrangian moduli space, and construct the developing map explicitly for a singularity corresponding to the type I n elliptic fiber (after hyper-Kähler rotation). Following Baues and Cortés, we show that various types of metric cones over two-dimensional ellip-tic affine spheres generate solutions of the Monge-Ampère equation in three dimensions. We then prove a local and global existence theorem for an elliptic affine two-sphere metric with prescribed singularities. The metric cone over the two-sphere minus three points yields a parabolic affine sphere with singularities along a " Y "-shaped locus. This gives a semi-flat Calabi-Yau metric in a neighborhood of the " Y " vertex.