Existence and non-existence of minimal graphic and $p$-harmonic functions

Jean-Baptiste Casteras (1, 2) , Esko Heinonen (3) , Ilkka Holopainen (3)

1 Département de mathématiques Université Libre de Bruxelles
2 MEPHYSTO-POST - Méthodes quantitatives pour les modèles aléatoires de la physique
3 Department of Mathematics and Statistics [Helsinki]

We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold $M$ with only one end if $M$ has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and $p$-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.

Domaines: Géométrie différentielle [math.DG]

Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen. Existence and non-existence of minimal graphic and $p$-harmonic functions. 2017. ⟨hal-01665512⟩