Cubic Differentials in the Differential Geometry of Surfaces

John Loftin; Ian McIntosh

Cubic Differentials in the Differential Geometry of Surfaces

Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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.

Original language	English
Title of host publication	Handbook of Teichmueller Theory
Editors	Athanase Papadopoulos
Publisher	European Mathematical Society
Number of pages	47
Volume	V
ISBN (Print)	978-3-03719-160-6
Publication status	Accepted/In press - 2015

Keywords

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

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Cite this

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AU - McIntosh, Ian

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AB - 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.

KW - math.DG

KW - math.GT

KW - 53C43 (primary), 20H10, 53A15 (secondary)

M3 - Chapter

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BT - Handbook of Teichmueller Theory

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PB - European Mathematical Society

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