**Title:** Degenerate Semi-Flat Calabi-Yau Metrics on S^2.

**Speaker:** Professor John Loftin

**Speaker Info:** Rutgers University Newark

**Brief Description:**

**Abstract:**

If $u$ is a convex function on a domain $\Omega\subset \mathbb{R}^n$ satisfying the Monge-Amp\`ere equation $\det(u_{ij}) = 1$, then there is a natural Calabi-Yau metric on the tube domain $\Omega + i \mathbb{R}^n$: Extend $u$ to be constant on the imaginary fibers and the K\"ahler metric $u_{i\bar\j}dz^i \overline{dz^j}$ is Ricci-flat. We call the real metric $u_{ij}dx^idx^j$ a semi-flat Calabi-Yau metric. Such metrics also may exist on manifolds which admit a flat affine connection $ abla$ on the tangent bundle and a volume form $\omega$ so that $ abla\omega=0$.Recently, such metrics have been of interest in understanding mirror symmetry, according to the conjecture of Strominger-Yau-Zaslow. In particular, Gross-Wilson have constructed many such metrics on $S2$ which are singular at 24 points, as real slices of limits of Calabi-Yau metrics on elliptic K3 surfaces. We construct many such metrics on $S2$, singular at any $n\ge6$ points, and compute the local affine structure near the singularities. The techniques involve affine differential geometry and a solving a semilinear PDE on $S2$ minus singularities.

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