Stable anisotropic minimal hypersurfaces in $\mathbf{R}^{4}$

TL;DR

This paper proves volume growth and bounds for stable anisotropic minimal hypersurfaces in four-dimensional space, providing explicit constants and an alternative proof of a Bernstein theorem using scalar curvature techniques.

Contribution

It introduces new volume estimates and an alternative proof for stable Bernstein theorem in $R^4$, connecting anisotropic minimal hypersurfaces with scalar curvature methods.

Findings

Complete, two-sided stable anisotropic minimal hypersurfaces have cubic volume growth.

Interior volume bounds are established for hypersurfaces in the unit ball.

Constants in volume estimates are explicitly calculated.

Abstract

We show that a complete, two-sided, stable immersed anisotropic minimal hypersurface in $\mathbf{R}^4$ has intrinsic cubic volume growth, provided the parametric elliptic integral is $C^2$-close to the area functional. We also obtain an interior volume upper bound for stable anisotropic minimal hypersurfaces in the unit ball. We can estimate the constants explicitly in all of our results. In particular, this paper gives an alternative proof of our recent stable Bernstein theorem for minimal hypersurfaces in $\mathbf{R}^4$. The new proof is more closely related to techniques from the study of strictly positive scalar curvature.