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

Nie, Xin; Seppi, Andrea

doi:10.2140/apde.2022.15.643

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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

DOI: 10.2140/apde.2022.15.643

Abstract

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.

Keywords

domain of dependence, affine differential geometry, affine Gauss–Kronecker curvature, Monge–Ampère equation

Mathematical Subject Classification 2010

Primary: 53A15

Secondary: 35J96, 53C42

Milestones

Received: 1 September 2019

Revised: 2 July 2020

Accepted: 13 November 2020

Published: 10 June 2022

Authors

Xin Nie
Shing-Tung Yau Center of Southeast University Southeast University Nanjing China

Andrea Seppi
CNRS and Université Grenoble Alpes Gières France

Vol. 15, No. 3, 2022