Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 127, 10 pages      arXiv:0712.4282      https://doi.org/10.3842/SIGMA.2007.127
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

On 1-Harmonic Functions

Shihshu Walter Wei
Department of Mathematics, The University of Oklahoma, Norman, Ok 73019-0315, USA

Received September 18, 2007, in final form December 17, 2007; Published online December 27, 2007

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 R; and every 7-dimensional SO(2) × SO(6)-invariant absolutely area-minimizing integral current in R8 is real analytic. The assumption on the SO(2) × SO(6)-invariance cannot be removed, due to the first counter-example in R8, proved by Bombieri, De Girogi and Giusti.

Key words: 1-harmonic function; 1-tension field; absolutely area-minimizing integral current.

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