Uniqueness of conformal-harmonic maps on locally conformally flat 4-manifolds

Motivated by the theory of harmonic maps on Riemannian surfaces, conformal-harmonic maps between two Riemannian manifolds $M$ and $N$ were introduced in search of a natural notion of harmonicity for maps defined on a general even dimensional Riemannian manifold $M$. They are critical points of a conformally invariant energy functional and reassemble the GJMS operators when the target is the set of real or complex numbers. On a four dimensional manifold, conformal-harmonic maps are the conformally invariant counterparts of the intrinsic bi-harmonic maps and a mapping version of the conformally invariant Paneitz operator for functions. In this paper, we consider conformal-harmonic maps from certain locally conformally flat 4-manifolds into spheres. We prove a quantitative uniqueness result for such conformal-harmonic maps as an immediate consequence of convexity for the conformally-invariant energy functional. To this end, we are led to prove a version of second order Hardy inequality on manifolds, which may be of independent interest.