A class of affine maximal surfaces with singularities and its relationship with minimal surface theory

TL;DR

This paper develops a global theory of affine maximal surfaces with singularities, introduces a special subclass called affine maxfaces, and explores their properties and connections to minimal surface theory, including inequalities and examples.

Contribution

It defines affine maxfaces as a new subclass of affine maximal surfaces with singularities and investigates their global properties and relationships to minimal surfaces.

Findings

Complete affine maxfaces satisfy an Osserman-type inequality.

Affine maxfaces do not include non-trivial improper affine fronts.

Examples relate affine maxfaces to Euclidean minimal surfaces.

Abstract

We study a global theory of affine maximal surfaces with singularities, which are called affine maximal maps and defined by Aledo--Mart\' inez--Mil\' an. In this paper, we define a special subclass of such surfaces other than improper affine fronts, called \emph{affine maxfaces}, and investigate their global properties with respect to certain notions of completeness. In particular, by applying Euclidean minimal surface theory, we show that ``complete'' affine maxfaces satisfy an Osserman-type inequality. Moreover, one can also observe that affine maxfaces are in a class that does not contain non-trivial improper affine fronts. We also provide examples of such surfaces which are related to Euclidean minimal surfaces.