The Equivariant Affine Minkowski Problem

The classical Minkowski problem asks for the construction of a strictly convex surface with specified Gaussian curvature and has already been extended to the Minkowski space by Bonsante and Fillastre. Through the use of affine differential geometry, one can generalize the Minkowski space to the setting of the affine space containing an open proper convex cone. That new setting gives birth to an equivariant affine Minkowski problem: can one introduce and fix the curvature of a convex surface invariant under the action of an affine deformation of a subgroup of SL(n,R) dividing our convex cone? I will show how that problem reduces to a Monge-Ampère equation and can be solved, generalising results from the work of Barbot, Béguin and Zeghib, of Bonsante and Fillastre, and of Nie and Seppi.

I will also explain how that work can be anchored in much bigger task of generalising the theory of flat Lorentzian spacetimes and its tools (causality, Cauchy surfaces and development, cosmological time, etc), which are linked to Teichmüller theory through the fundamental work of Mess, to a theory of affine spacetimes then linked to higher Teichmüller theory.