Affine Spheres and Kähler-Einstein Metrics

Abstract

In this note, we introduce a straightforward correspondence between some natural affine Kahler metrics on convex cones and natural metrics on certain hypersurfaces asymptotic to the boundary of these cones. Recall an affine Kahler metric is a Riemannian metric locally given by the Hessian of a potential function φ, i.e. gij = ∂2φ ∂xi∂xj . Note that this metric is well defined only up to affine coordinate changes. Affine Kahler metrics are sometimes called Hessian metrics. See e.g. [4, 12]. The centroaffine second fundamental form provides a Riemannian metric on a hypersurface H ⊂ R, if H is the radial graph of a function − 1 u , with u negative and convex. The formula for this metric is − 1 u ∂2u ∂ti∂tj , and if u transforms as section of a certain line bundle, this metric makes sense under projective coordinate changes. See e.g. [10]. Form a cone