A $C^2$-Perturbative Bernstein Theorem for Anisotropic Entire Minimal Graphs

Abstract

We prove a Bernstein theorem for $\Phi$-anisotropic minimal hypersurfaces in dimensions $1\le n\le 7$ that the only entire smooth solutions $u$ of $\Phi$-anisotropic minimal hypersurfaces equation are affine functions provided the anisotropic area functional integrand $\Phi$ is sufficiently $C^2$--close to the Euclidean area integrand. This settles the $C^2$ entire--graph version of anisotropic Bernstein problem posed by Mooney and Yang \cite{MooneyYang2024}, and the proof uses a compactness--rigidity argument combined with Figalli's regularity theorem established in \cite{Figalli2017}.

We solve the $C^2$ entire-graph version of the anisotropic Bernstein problem posed by Mooney-Yang \cite{MooneyYang2024} by proving that, for $1\le n\le 7$, entire smooth solutions $u$ of $\Phi$-anisotropic minimal hypersurfaces equation are linear functions provided the anisotropic integrand is sufficiently $C^2$--close to the classical area functional