Algebraic zero mean curvature varieties in semi-riemannian manifolds

TL;DR

This paper constructs and classifies algebraic space-like surfaces with zero mean curvature in anti de Sitter and de Sitter spaces, revealing the existence of algebraic maximal hypersurfaces of any order in these Lorentzian manifolds.

Contribution

It introduces a family of algebraic space-like surfaces in anti de Sitter space and classifies all order two algebraic maximal hypersurfaces, expanding understanding of zero mean curvature varieties.

Findings

Existence of algebraic maximal examples of any order in anti de Sitter space

Complete classification of order two algebraic maximal hypersurfaces in anti de Sitter space

Examples of algebraic zero mean curvature hypersurfaces in de Sitter space

Abstract

In this paper we provide a family of algebraic space-like surfaces in the three dimensional anti de Sitter space that shows that this Lorentzian manifold admits algebraic maximal examples of any order. Then, we classify all the space-like order two algebraic maximal hypersurfaces in the anti de Sitter $N$-dimensional space. Finally, we provide two families of examples of Lorentzian order two algebraic zero mean curvature in the de Sitter space.