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arXiv · 2507.10035

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

Jun Matsumoto
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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.

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Jun Matsumoto. 2025-07-14. A class of affine maximal surfaces with singularities and its relationship with minimal surface theory. https://arxiv.org/abs/2507.10035

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