The Journal of Geometric Analysis Volume 9, Number 4, 1999 Sharp Stability Results for Almost Conformal Maps in Even Dimensions By Stefan Miiller, Vladimir Sverdk, and Baisheng Yan ABSTRACT. 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 + Imln/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. 1. Introduction Let n > 2 and ~2 be a domain in R n. We denote by WI'P(f2; R n) (p > 1) the usual space of all Sobolev maps u: g2 ~ R n . A map u c WI'p(f2; R n) is called conformal if Vu(x)~R +SO(n)={kQ]L>0, QcSO(n)} a.e.x ~f2. Here, R + denotes all nonnegative real numbers, and SO(n) denotes the set of all rotations with determinant equal to 1. A classical Liouville's theorem asserts that if n > 3 and p > n, then a conformal map in WI'p(f2; R n) must be a restriction onto ~2 of a M6bius map (see [4] and [26]). A recent result of Iwaniec and Martin [16] shows that in even dimensions Liouville's theorem is still true for conformal maps in W I'p if p > n/2. In odd dimensions, Liouville's theorem holds for conformal maps in W l'p if p is not too far below n; the minimal value of all such pts is unknown (see [14] and [17]). Note that there are counterexamples in all dimensions showing that a conformal map in W I'p for p < n/2 may not be a restriction of a M6bius map (see, e.g., [16]). In this paper, we are mainly interested in the stability of conformal maps, i.e., the question whether the weak limit of almost conformal maps is conformal. In the following, weak convergence is denoted by the half-arrow "--~" and strong convergence by the arrow "--+." Our main result is the following: Theorem 1.1. Suppose n _> 4 / s e v e n and that {uJ} is a sequence in wl'n/2(~'2; R n) and satisfies u j-~u in W l'n/2(f2;R n), (1.1) Math Subject Classifications. 30C65, 35A15. Key Words and Phrases. quasi-conformal mappings, stability, quasi-convex functions. 91999 The Journal of Geometric Analysis ISSN1050-6926