## Illinois Journal of Mathematics

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

Ye-Lin Ou

#### Abstract

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.

#### Article information

Source
Illinois J. Math., Volume 49, Number 3 (2005), 911-927.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138228

Digital Object Identifier
doi:10.1215/ijm/1258138228

Mathematical Reviews number (MathSciNet)
MR2210268

Zentralblatt MATH identifier
1089.58010

#### Citation

Ou, Ye-Lin. $p$-harmonic functions and the minimal graph equation in a Riemannian manifold. Illinois J. Math. 49 (2005), no. 3, 911--927. doi:10.1215/ijm/1258138228. https://projecteuclid.org/euclid.ijm/1258138228