On 1-Harmonic Functions

Details On 1-Harmonic Functions On 1-Harmonic Functions

2007-12-27

arxiv.org/abs/0712.4282v1

SIGMA 3 (2007), 127, 10 pages

Mathematics

Differential Geometry

This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in

Scientific paper

10.3842/SIGMA.2007.127

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.

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