On 1-Harmonic Functions

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.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA