# Affine spheres with prescribed Blaschke metric

## TL;DR

This paper establishes a precise mathematical condition involving Gaussian curvature that characterizes when a 2D metric can be realized as the Blaschke metric of an affine sphere, linking curvature properties to geometric realizability.

## Contribution

It proves that a specific curvature condition is both necessary and sufficient for a metric to be realized as an affine sphere's Blaschke metric, providing a complete characterization.

## Key findings

- The curvature condition $ riangle ext{ln}| ext{ extkappa}- extlambda|=6 ext{ extkappa}$ is equivalent to the metric's realizability.
- The result offers a criterion for identifying metrics that correspond to affine spheres.
- The proof connects differential equations with geometric realizability in affine differential geometry.

## Abstract

It is proved that the equality $\Delta\ln|\kappa-\lambda|=6\kappa$, where $\kappa$ is the Gaussian curvature of a metric tensor g on a 2-dimensional manifold is a sufficient and necessary condition for local realizability of the metric as the Blaschke metric of some affine sphere.