On 1-Harmonic Functions

TL;DR

This paper characterizes entire subsolutions of the 1-harmonic equation with applications in geometry, showing level hypersurfaces are area-minimizing and establishing analyticity of certain invariant minimal currents in high dimensions.

Contribution

It provides new characterizations of 1-harmonic subsolutions and demonstrates that invariant minimal currents are real analytic, with counterexamples highlighting the importance of symmetry assumptions.

Findings

Level hypersurfaces are calibrated and area-minimizing.

7D invariant minimal currents are real analytic.

Symmetry assumptions are essential for the results.

Abstract

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1$-tension field are given with applications in geometry via transformation group theory. In particular, we prove that every level hypersurface of such a subsolution is calibrated and hence is area-minimizing over $\mathbb{R}$; and every 7-dimensional $SO(2)\times SO(6)$-invariant absolutely area-minimizing integral current in $\mathbb{R}^8$ is real analytic. The assumption on the $SO(2) \times SO(6)$-invariance cannot be removed, due to the first counter-example in $\mathbb{R}^8$, proved by Bombieri, De Girogi and Giusti.