Energy convexity of intrinsic bi-harmonic maps and applications I: Spherical target

In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball {B_{1}\subset\mathbb{R}^{4}} into the sphere {\mathbb{S}^{n}}. In particular, this yields a version of uniqueness of weakly harmonic maps on the unit 4-ball which is new. We also show a version of energy convexity along the intrinsic bi-harmonic map heat flow into {\mathbb{S}^{n}}, which in particular yields the long-time existence of the intrinsic bi-harmonic map heat flow, a result that was until now only known assuming the non-positivity of the target manifolds by Lamm [26]. Further, we establish the previously unknown result that the energy convexity along the flow yields uniform convergence of the flow.