Sharp stability results for almost conformal maps in even dimensions

Let ~ C R n and n > 4 be even. We show that if a sequence {uJ } in W l'n/2 (f2; R n) is almost conformal in the sense that dist (Vu j , R+ SO(n) ) converges strongly to 0 in L n/2 and if u j converges weakly to u in W 1,n/2, then u is conformal and VuJ ~ Vu strongly in Lqoc for all 1 < q < n/2. It is known that this conclusion fails if n~2 is replaced by any smaller exponent p. We also prove the existence of a quasiconvex function f ( A ) that satisfies 0 < f ( A ) < C (1 + Iml n/2) and vanishes exactly on R + SO(n). The proof of these results involves the lwaniec-Martin characterization of conforrnal maps, the weak continuity and biting convergence of Jacobians, and the weak-L 1 estimates for Hodge decompositions.