Cubic Differentials in the Differential Geometry of Surfaces

John Loftin, Ian McIntosh

Research output: Chapter in Book/Report/Conference proceedingChapter


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.
Original languageEnglish
Title of host publicationHandbook of Teichmueller Theory
EditorsAthanase Papadopoulos
PublisherEuropean Mathematical Society
Number of pages47
ISBN (Print)978-3-03719-160-6
Publication statusAccepted/In press - 2015


  • math.DG
  • math.GT
  • 53C43 (primary), 20H10, 53A15 (secondary)

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