Conformal change of Riemannian metrics and biharmonic maps

TL;DR

This paper investigates the existence of solutions to a differential equation related to biharmonic maps under conformal metric changes, revealing dimension-dependent behaviors and constructing new metrics with specific harmonic map properties.

Contribution

It demonstrates the nonexistence of global positive solutions for certain dimensions and constructs new conformal metrics on 3-manifolds that alter harmonic map properties.

Findings

No global positive solutions for dimensions m ≥ 5.

Existence of global positive solutions when m=3.

Construction of conformal metrics on 3-manifolds affecting harmonic maps.

Abstract

For the reduction ordinary differential equation due to Baird and Kamissoko \cite{BK} for biharmonic maps from a Riemannian manifold $(M^m,g)$ into another one $(N^n,h)$, we show that this ODE has no global positive solution for every $m\geq 5$. On the contrary, we show that there exist global positive solutions in the case $m=3$. As applications, for the the Riemannian product $(M^3,g)$ of the line and a Riemann surface, we construct the new metric $\widetilde{g}$ on $M^3$ conformal to $g$ such that every nontrivial product harmonic map from $M^3$ with respect to the original metric $g$ must be biharmonic but not harmonic with respect to the new metric $\widetilde{g}$.