IUMJ

Title: Conformal change of Riemannian metrics and biharmonic maps

Authors: Hisashi Naito and Hajime Urakawa

Issue: Volume 63 (2014), Issue 6, 1631-1657

Abstract:

For the reduction ordinary differential equation due to Baird and Kamissoko [P. Baird and D. Kamissoko, \textit{On constructing biharmonic maps and metrics}, Ann. Global Anal. Geom. \textbf{23} (2003), no. 1, 65--75] 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 $\widetilde{g}$.