  We study the real Monge-Amp\`ere 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 holomorphic 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-K\"ahler rotation). Following Baues and Cort\'es, we show that various
types of metric cones over two-dimensional elliptic affine spheres generate
solutions of the Monge-Amp\`ere 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.
  In the erratum, we correct a gap in the solution of the elliptic affine
sphere equation (with the small caveat that a parameter in the equation, the
cubic differential, must be smaller than we assumed before). The new proof
still allows us to construct semi-flat Calabi-Yau metrics in a neighborhood of
the "Y" vertex by appealing to the result of Baues-Cort\'es. We also give an
alternate construction of such semi-flat Calabi-Yau metrics by using a result
on hyperbolic affine spheres due to the first author.
