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

Abstract

Abstract In this paper, we show an energy convexity and thus uniqueness for weakly intrinsic bi-harmonic maps from the unit 4-ball B 1 ⊂ ℝ 4 {B_{1}\subset\mathbb{R}^{4}} into the sphere 𝕊 n {\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 𝕊 n {\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.