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

2025/07/14 by Jun Matsumoto, Matsumoto, Jun
Mathematics · #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2507.10035

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 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.

Citations

• Bloch-Ros principle and its application to surface theory
• On the maximal number of exceptional values of Gauss maps for various\n classes of surfaces
• Value distribution of the Gauss map of improper affine spheres
• A representation formula for indefinite improper affine spheres
• Singularities of Maximal Surfaces
• The geometry of fronts
• Spacelike CMC 1 surfaces with elliptic ends in de Sitter 3-Space
• Maximal surfaces with singularities in Minkowski space
• Flat fronts in hyperbolic 3-space
• Minimal surfaces that attain equality in the Chern-Osserman inequality

Related

• Maximal surfaces with singularities in Minkowski space
• Affine maximal hypersurfaces
• Calabi affine maximal surfaces and centroaffine Bernstein problems
• Singular Points of Affine ML-Surfaces
• On the hypersufaces of the Euclidean space which are simultaneously\n minimal and maximal