Some calibrated surfaces in manifolds with density

TL;DR

This paper demonstrates that certain classical surfaces like hyperplanes, hyperspheres, and hypercylinders are weighted area-minimizing in manifolds with density using calibration techniques, and extends this to local minimality of weighted hypersurfaces.

Contribution

The paper introduces calibration methods to prove weighted minimality of classical surfaces and extends minimality results to weighted hypersurfaces in manifolds with density.

Findings

Hyperplanes, hyperspheres, and hypercylinders are weighted minimizing surfaces.

Calibration methods effectively prove weighted minimality.

Weighted minimal hypersurfaces are locally area-minimizing.

Abstract

Hyperplanes, hyperspheres and hypercylinders in $\Bbb R^n$ with suitable densities are proved to be weighted minimizing by a calibration argument. Also calibration method is used to prove a weighted minimal hypersurface is weighted area-minimizing locally.