A class of affine maximal surfaces with singularities and its relationship with minimal surface theory
Differential Geometry
2025-07-15 v1
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.
Keywords
Cite
@article{arxiv.2507.10035,
title = {A class of affine maximal surfaces with singularities and its relationship with minimal surface theory},
author = {Jun Matsumoto},
journal= {arXiv preprint arXiv:2507.10035},
year = {2025}
}