Sharp stability results for almost conformal maps in even dimensions

Let Ω ⊂ Rⁿ and n ≥ 4 be even. We show that if a sequence {u^j} in W^1,n/2(Ω Rⁿ) is almost conformal in the sense that dist (∇u^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 ∇u^j → ∇u strongly in L_loc^q 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 + |A|^n/2) and vanishes exactly on R⁺ SO(n). The proof of these results involves the Iwaniec-Martin characterization of conformal maps, the weak continuity and biting convergence of Jacobians, and the weak-L¹ estimates for Hodge decompositions.