The Riemannian hemisphere is almost calibrated in the injective hull of its boundary

TL;DR

This paper constructs a special differential form in the injective hull of a circle that reveals a nearly optimal calibration property of the Riemannian hemisphere, providing new bounds relevant to Gromov's filling area conjecture.

Contribution

It introduces a differential form with stationary comass norm in the injective hull of a circle, linking geometric measure theory with the filling area conjecture.

Findings

The form is stationary on the open hemisphere.

Provides a lower bound for the Finsler mass of certain currents.

Supports Gromov's filling area conjecture.

Abstract

An exact differential two-form is constructed in the injective hull of the Riemannian circle, whose comass norm, defined via the inscribed Riemannian area on normed planes, is stationary at every point of the open hemisphere spanned by the circle. As a consequence, in any metric space, the induced Finsler mass of a two-dimensional Ambrosio-Kirchheim rectifiable current with boundary a Riemannian circle of length $2\pi$ admits a lower bound of $2\pi$ plus a second-order term in the Hausdorff distance to an isometric copy of the hemisphere. This estimate applies to all oriented Lipschitz surfaces spanning the circle, regardless of their topology, and thus offers positive evidence for Gromov's filling area conjecture.