Vol. 15, No. 3, 2022

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Regular domains and surfaces of constant Gaussian curvature in 3-dimensional affine space

Xin Nie and Andrea Seppi

Vol. 15 (2022), No. 3, 643–697

Generalizing the notion of domains of dependence in the Minkowski space, we define and study regular domains in the affine space with respect to a proper convex cone. In three dimensions, we show that every proper regular domain is uniquely foliated by some particular surfaces with constant affine Gaussian curvature. The result is based on the analysis of a Monge–Ampère equation with extended real-valued lower semicontinuous boundary condition.

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domain of dependence, affine differential geometry, affine Gauss–Kronecker curvature, Monge–Ampère equation
Mathematical Subject Classification 2010
Primary: 53A15
Secondary: 35J96, 53C42
Received: 1 September 2019
Revised: 2 July 2020
Accepted: 13 November 2020
Published: 10 June 2022
Xin Nie
Shing-Tung Yau Center of Southeast University
Southeast University
Andrea Seppi
CNRS and Université Grenoble Alpes