Polygonal surfaces in pseudo-hyperbolic spaces

Abstract

A polygonal surface in the pseudo-hyperbolic space $H^{2,n}$ is a complete maximal surface bounded by a lightlike polygon in the Einstein universe ${\mathrm{Ein}}^{1,n}$ with finitely many vertices. In this article, we give several characterizations of them. Polygonal surfaces are characterized by finiteness of their total curvature and by asymptotic flatness. They have parabolic type and polynomial quartic differential. Our result relies on a comparison between three ideal boundaries associated with a maximal surface, corresponding to three distinct distances naturally defined on the maximal surface.