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

TL;DR

This paper investigates conditions under which minimal graph and p-harmonic functions are constant or non-constant on certain Riemannian manifolds, providing both non-existence and existence results based on curvature assumptions.

Contribution

It establishes new non-existence results for bounded solutions on manifolds with non-negative curvature and constructs explicit examples of solutions on symmetric manifolds.

Findings

Bounded minimal solutions are constant on manifolds with one end and non-negative curvature.

Existence of bounded non-constant solutions on rotationally symmetric Cartan-Hadamard manifolds.

Optimal curvature conditions for existence and non-existence of solutions.

Abstract

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.