$p$-harmonic functions and the minimal graph equation in a Riemannian manifold

Ou, Ye-Lin

Ou, Ye-Lin

Illinois J. Math., Tome 49 (2005) no. 2, p. 911-927 / Harvested from Project Euclid

Résumé

We study the minimal graph equation in a Riemannian manifold. After explaining the geometric meaning of the solutions and giving some entire solutions of the minimal graph equation in Nil space and in a hyperbolic space we find a link among $p$-harmonicity, horizontal homothety, and the minimality of the vertical graphs of a submersion. We also study the transformation of the minimal graph equation under the conformal change of metrics. We prove that the foliation by the level hypersurfaces of a $p$-harmonic submersion is a minimal foliation with respect to a conformally deformed metric. This implies, in particular, that the graph of any harmonic function from a Euclidean space is a minimal hypersurface in a complete conformally flat space, thus providing an effective way to construct (foliations by) minimal hypersurfaces.

Publié le : 2005-07-15
Classification: 58E20, 49Q05, 53C12

@article{1258138228,
     author = {Ou, Ye-Lin},
     title = {$p$-harmonic functions and the minimal graph equation in a Riemannian manifold},
     journal = {Illinois J. Math.},
     volume = {49},
     number = {2},
     year = {2005},
     pages = { 911-927},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1258138228}
}

Ou, Ye-Lin. $p$-harmonic functions and the minimal graph equation in a Riemannian manifold. Illinois J. Math., Tome 49 (2005) no. 2, pp.  911-927. http://gdmltest.u-ga.fr/item/1258138228/