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Notes on a paper of Mess
These notes are a companion to the article "Lorentz spacetimes of constant
curvature" by Geoffrey Mess, which was first written in 1990 but never
published. Mess' paper will appear together with these notes in a forthcoming
issue of Geometriae Dedicata.Comment: 26 page
Cubic Differentials in the Differential Geometry of Surfaces
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 . We also discuss the global theory and applications to
representations of surface groups and to mirror symmetry.Comment: corrected published editio
Lengthening deformations of singular hyperbolic tori
Let be a torus with a hyperbolic metric admitting one puncture or cone
singularity. We describe which infinitesimal deformations of lengthen (or
shrink) all closed geodesics. We also study how the answer degenerates when
becomes Euclidean, i.e. very small.Comment: 17 pages, 5 figures. To appear in: Annales de la facult\'e des
sciences de Toulouse (volume dedicated to Michel Boileau's sixtieth birthday
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