Cubic Differentials in the Differential Geometry of Surfaces

TL;DR

This paper explores the differential geometry of convex affine spheres and minimal Lagrangian surfaces, revealing their structure through holomorphic cubic differentials and connections to affine Toda systems, with implications for surface group representations and mirror symmetry.

Contribution

It introduces a unified framework linking the local geometry of these surfaces with affine Toda systems and discusses global applications in representation theory and mirror symmetry.

Findings

Holomorphic cubic differentials characterize surface geometry.

Affine Toda systems describe structure equations.

Applications to surface group representations and mirror symmetry.

Abstract

We discuss the local differential geometry of convex affine spheres in $\re^3$ and of minimal Lagrangian surfaces in Hermitian symmetric spaces. In each case, there is a natural metric and cubic differential holomorphic with respect to the induced conformal structure: these data come from the Blaschke metric and Pick form for the affine spheres and from the induced metric and second fundamental form for the minimal Lagrangian surfaces. The local geometry, at least for main cases of interest, induces a natural frame whose structure equations arise from the affine Toda system for $\mathfrak a^{(2)}_2$. We also discuss the global theory and applications to representations of surface groups and to mirror symmetry.