Calabi affine maximal surfaces and centroaffine Bernstein problems

TL;DR

This paper investigates Calabi affine maximal surfaces and centroaffine Bernstein problems, providing classifications, new examples, and solving all five Bernstein problems posed in 2004.

Contribution

It proves that Calabi extremal surfaces are also maximal in Calabi affine geometry and solves all five Bernstein problems for centroaffine extremal hypersurfaces.

Findings

Classified special Calabi affine maximal surfaces

Constructed new complete Calabi affine maximal surfaces

Solved all five Bernstein problems for centroaffine extremal hypersurfaces

Abstract

Motivated by Calabi's calculation of the second variation sign for locally strongly convex affine maximal surfaces in equiaffine geometry, we first prove that every Calabi extremal surface is also maximal in the Calabi affine geometry. By employing suitably chosen orthonormal frame fields and analyzing the corresponding Codazzi equations, we then obtain local classifications for certain special classes of Calabi affine maximal surfaces and hyperbolic centroaffine extremal surfaces. These examples inspire the construction of new, complete Calabi affine maximal surfaces and centroaffine extremal hypersurfaces. Notably, the complete centroaffine extremal hypersurfaces we establish answer all five centroaffine Bernstein problems posed by Li- Li-Simon in 2004.