On 1-Harmonic Functions

Shihshu Walter Wei

doi:10.3842/SIGMA.2007.127

paper

On 1-Harmonic Functions

arXiv:0712.4282 · doi:10.3842/SIGMA.2007.127

Abstract

Characterizations of entire subsolutions for the 1-harmonic equation of a constant 1 $- t e n s i o n f i e l d a r e g i v e n w i t ha ppl i c a t i o n s in g eo m e t r y v ia t r an s f or ma t i o n g r o u pt h eor y . I n p a r t i c u l a r, w e p r o v e t ha t e v er y l e v e l h y p er s u r f a ceo f s u c ha s u b so l u t i o ni sc a l ib r a t e d an d h e n ce i s a r e a - minimi z in g o v er$ \mathbb{R} $; an d e v er y 7 - d im e n s i o na l$ SO(2)\times SO(6) $- in v a r ian t ab so l u t e l y a r e a - minimi z in g in t e g r a l c u r r e n t in$ \mathbb{R}^8 $i sr e a l ana l y t i c . T h e a ss u m pt i o n o n t h e$ SO(2) \times SO(6) $- in v a r ian cec ann o t b er e m o v e d, d u e t o t h e f i r s t co u n t er - e x am pl e in$ \mathbb{R}^8$, proved by Bombieri, De Girogi and Giusti.

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/