Calabi affine maximal surfaces and centroaffine Bernstein problems

TL;DR

This work connects Calabi affine maximal surface theory with centroaffine Bernstein problems by proving that Calabi extremal surfaces are Calabi affine maximal, and by deriving frame-based classifications and Codazzi relations that yield new complete examples. It leverages the Calabi metric, the Euler–Lagrange equation $\Delta\ln\det(f_{ij})=0$, and the Tchebychev vector $T$ to classify special Calabi affine maximal surfaces and hyperbolic centroaffine extremal surfaces, and then constructs complete centroaffine extremal hypersurfaces via Calabi products. The authors produce explicit models, including flat Calabi-affine maximal surfaces with complete Calabi metrics and complete elliptic centroaffine hypersurfaces with nonconstant $|T|$, and a rich family of complete hyperbolic centroaffine extremal surfaces with $K=-1$ that solve several Bernstein problems. Collectively, the results deliver complete answers to Bernstein problems II–V, provide counterexamples to I and III, and highlight a deep connection between Calabi affine and centroaffine geometries with concrete, computable examples. These findings advance understanding of Bernstein-type questions in affine and centroaffine differential geometry and expand the catalog of complete, nontrivial 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.

Figures (4)

• Figure 1: The graph of (\ref{['5.1']}).
• Figure 2: The graph of Theorem 6.2 (i).
• Figure 3: The graph of Theorem 6.2 (ii).
• Figure 4: The graph of Theorem 6.2 (v).