Cones over minimal products cannot be calibrated by smooth calibrations

TL;DR

This paper proves that cones over minimal product structures cannot be calibrated by any smooth calibration in Euclidean spaces, extending previous results on minimal submanifolds.

Contribution

It establishes a fundamental obstruction caused by minimal product structures preventing cones over such products from being calibrated by smooth calibrations.

Findings

Cones over non-trivial minimal products are not calibratable by smooth calibrations.

The result extends previous work by Zha26 on calibration obstructions.

Minimal product structures inherently obstruct the existence of global smooth calibrations.

Abstract

We extend a key result in [Zha26], by establishing the obstruction that the minimal product structure (for minimal submanifolds or stationary currents in spheres) automatically makes all cones over (non-trivial) minimal products fail to be calibrated by any global defined smooth calibration in Euclidean spaces.