Global existence and uniqueness of Schrödinger maps in dimensions d⩾4d⩾4

In dimensions d⩾4d⩾4, we prove that the Schrödinger map initial-value problem{∂ts=s×ΔsonRd×R;s(0)=s0 admits a unique solution s:Rd×R→S2↪R3, s∈C(R:HQ∞), provided that s0∈HQ∞ and ‖s0−Q‖H˙d/2≪1, where Q∈S2Q∈S2.