Stability of vertical and radial graphs in the Euclidean space with density

TL;DR

This paper proves the strong stability of vertical and radial graphs in Euclidean space with specific densities, including cases like translators and expanders, and shows stationary vertical graphs are weighted minimizers under certain conditions.

Contribution

It introduces new stability results for vertical and radial graphs in Euclidean space with densities, using calibration techniques to identify weighted minimizers.

Findings

Vertical and radial graphs are strongly stable for certain densities.

Stationary vertical graphs are weighted minimizers when densities depend on a spatial coordinate.

Results include special cases like translators, expanders, and singular minimal hypersurfaces.

Abstract

It is proved that vertical graphs and radial graphs are strongly stable for a certain type of densities in Euclidean space ${\mathbb R}^{n+1}$. Particular cases of these densities include translators, expanders and singular minimal hypersurfaces. Using techniques of calibrations, it is also proved that for densities depending on a spatial coordinate, stationary vertical graphs are weighted minimizers in a certain class of hypersurfaces.