Stable anisotropic minimal hypersurfaces in $\mathbb{R}^{5}$ and $\mathbb{R}^{6}$

TL;DR

This paper resolves the stable anisotropic Bernstein problem in dimensions corresponding to $\mathbb{R}^5$ and $\mathbb{R}^6$ by proving that complete, two-sided stable $F$-minimal hypersurfaces with $\mathcal{A}_F$ sufficiently close to the area functional are flat. The authors combine a conformal deformation $\tilde{g}=r^{-2}g$ with a carefully engineered $\alpha$-bi-Ricci curvature spectral bound, then implement a warped $\mu$-bubble framework to obtain sharp volume-growth controls via a weighted area functional. Key steps include establishing one-endness under pinching, translating stability into a positive spectral bound on $\tilde{\mathrm{biRic}}_\alpha$, and constructing warped $\mu$-bubbles to derive explicit volume bounds and polynomial growth for geodesic balls. The work yields explicit constants for $n=4,5$ and provides a robust pathway to flatness results in the anisotropic setting, extending prior Bernstein-type results to the closest-anisotropic regime. Overall, the paper advances the understanding of when anisotropic minimal hypersurfaces must be flat under small perturbations of the area functional, with potential implications for geometric analysis and variational problems in higher codimension.

Abstract

In this paper, we prove that a complete, two-sided, stable anisotropic minimal immersed hypersurface in $\mathbb{R}^{5}$ or $\mathbb{R}^{6}$ is flat, provided the anisotropic area functional is $C^4$-close to the area functional.

Theorems & Definitions (23)

• Theorem 1: Chodosh-Li-anisotropic
• Theorem 2
• Corollary 3
• Proposition 4: Chodosh-Li-anisotropic
• Proposition 5: Chodosh-Li-anisotropic
• Proposition 6: Chodosh-Li-anisotropic
• Lemma 7
• proof
• Lemma 8
• proof