On p-harmonic maps and convex functions

TL;DR

The paper demonstrates that, generally, the composition of a p-harmonic map with a convex function is not p-subharmonic, and it explores conditions under which this property may or may not hold.

Contribution

It provides a general proof that the composition of a p-harmonic map with a convex function is not p-subharmonic, and introduces symmetry assumptions to analyze the problem via differential inequalities.

Findings

Composition of p-harmonic maps and convex functions is not p-subharmonic in general.

Symmetry assumptions reduce the problem to an ordinary differential inequality.

Asymptotic estimates are key to understanding the behavior of p-harmonic maps.

Abstract

We prove that, in general, given a $p$-harmonic map $F:M\to N$ and a convex function $H:N\to\mathbb{R}$, the composition $H\circ F$ is not $p$-subharmonic. By assuming some rotational symmetry on manifolds and functions, we reduce the problem to an ordinary differential inequality. The key of the proof is an asymptotic estimate for the $p$-harmonic map under suitable assumptions on the manifolds.