A sharp Bernstein theorem for affine maximal type surfaces

Abstract

We investigate Bernstein theorem for the affine maximal type equation \begin{equation*} \sum_{i,j=1}^nf^{ij}(D^a)_{ij}=0,\qquad D=\det(f_{ij}),\qquad a\neq0, \end{equation*} under completeness of the Calabi metric. In dimension two, we prove that every smooth strictly convex solution on a convex domain is a quadratic polynomial whenever $a\notin[-\tfrac23,-\tfrac13]$, resolving a conjecture of Sun and Xu [25]. For every $n\geq2$, we also construct explicit non-quadratic Calabi complete solutions on convex domains for all $a\in[-\tfrac n{n+1},-\tfrac1{n+1}]$. These examples show that the rigidity range is sharp in dimension two.