Rigidity estimates for isometric and conformal maps from $\mathbb{S}^{n-1}$ to $\mathbb{R}^n$

We investigate both linear and nonlinear stability aspects of rigid motions (resp. M\"obius transformations) of $\mathbb{S}^{n-1}$ among Sobolev maps from $\mathbb{S}^{n-1}$ into $\mathbb{R}^n$. Unlike similar in flavour results for maps defined on domains of $\mathbb{R}^n$ and mapping into $\mathbb{R}^n$, not only an isometric (resp. conformal) deficit is necessary in this more flexible setting, but also a deficit measuring the distortion of $\mathbb{S}^{n-1}$ under the maps in consideration. The latter is defined as an associated isoperimetric type of deficit. We mostly focus on the case $n=3$, where we also explain why the estimates are optimal in their corresponding settings. In the isometric case the estimate holds true also when $n=2$ and generalizes in dimensions $n\geq 4$ as well, if one requires apriori boundedness in a certain higher Sobolev norm. We also obtain linear stability estimates for both cases in all dimensions. These can be regarded as Korn-type inequalities for the combination of the quadratic form associated with the isometric (resp. conformal) deficit on $\mathbb{S}^{n-1}$ and the isoperimetric one.


Introduction
In this paper we examine stability issues of isometric and conformal maps from S n−1 into R n of relatively low regularity, focusing mostly, but not solely, on the case n = 3. Since the starting domain is of codimension 1 in R n , these maps exhibit of course more flexibility than their analogues from open subdomains of R n into R n . On the one hand, isometric and conformal maps are actually rigid when considered from S n−1 into itself, as the following version of the well known theorem by J. Liouville asserts. Theorem 1.1. (Liouville's Theorem on S n−1 ) (i) Let n ≥ 2 and p ∈ [1, +∞]. A generalized orientation-preserving (\-reversing) u ∈ W 1,p (S n−1 ; S n−1 ) is isometric iff it is a rigid motion of S n−1 , i.e., iff there exists O ∈ O(n) so that for every x ∈ S n−1 , (1.1) (ii) Let n ≥ 3. A generalized orientation-preserving (\-reversing) u ∈ W 1,n−1 (S n−1 ; S n−1 ) of degree 1 (\-1) is conformal iff it is a Möbius transformation of S n−1 , i.e., iff there exist O ∈ O(n), ξ ∈ S n−1 and λ > 0 so that for every x ∈ S n−1 , Here, φ ξ,λ := σ −1 ξ • i λ • σ ξ , where σ ξ is the stereographic projection of S n−1 onto T ξ S n−1 ∪ {∞}, and i λ : T ξ S n−1 → T ξ S n−1 is the dilation in T ξ S n−1 by factor λ > 0.
On the other hand however, there is a wide variety of such maps from S n−1 into R n . In contrast to the classical rigidity in the Weyl problem for isometric embeddings, according to which the only C 2 (or even C 1,α for α > 2 3 ) isometric embedding of S n−1 into R n is the standard one modulo rigid motions (cf. [3], [8], [9], [14]), as a consequence of the celebrated Nash-Kuiper theorem (cf. [16], [20]), the following paradox happens for less regular, say C 1 isometric embeddings.
Given any δ ∈ (0, 1), in an arbitrarily small C 0 -neighbourhood of the short homothety u δ : S n−1 → R n , u δ (x) := δx, there exist C 1 isometric embeddings, which can be visualized as wrinkling isometrically S n−1 inside the small ball B δ (0) in a way that produces continuously changing tangent planes. For the more general case of conformal maps from S n−1 to R n , at least when n = 3, other examples that are not Möbius transformations are provided by the Uniformization Theorem and some of them have often been used in cartography, for instance the inverse of Jacobi's conformal map projection that smoothly and conformally maps S 2 onto the surface of an ellipsoid. Therefore, Liouville's rigidity theorem on S n−1 on the one hand, and the aforementioned flexibility phenomena on the other, indicate the following fact. When one seeks stability of the isometry (resp. the conformal) group of S n−1 among Sobolev maps u : S n−1 → R n , apart from an isometric (resp. conformal ) deficit, an extra deficit measuring the deviation of u(S n−1 ) from being a round sphere is necessary. In this paper we make a connection between stability aspects for these two classes of mappings and the isoperimetric inequality, and this extra deficit should be interpreted in both cases as an isoperimetric type of deficit produced by the maps in consideration.
With the notations that we adopt in Section 2, our main result in the isometric case is the following. Theorem 1.2. There exists c 1 > 0 so that for every u ∈ W 1,2 (S 2 ;

3)
where 0 ≤ σ 1 ≤ σ 2 are the principal stretches of u, i.e., the eigenvalues of √ ∇ T u t ∇ T u, and The first term on the right hand side of (1.3) is an L 2 -isometric deficit of u penalizing local stretches, while the second term (in the definition of which in (1.4) we use the identification between a 2-simple vector and its Hodge dual) represents in this setting the isoperimetric deficit of u. Since isometric maps preserve the surface area of S 2 , the latter reduces in this situtation to the positive part of the excess in the signed volume produced by u. The exact analogue of Theorem 1.2 holds true also in dimension n = 2 (see Proposition 3.2 in Section 3) and, as long as u satisfies an apriori bound on its homogeneous W 1,2(n−2) -seminorm, also in dimensions n ≥ 4, as stated in the following. where 0 ≤ σ 1 ≤ · · · ≤ σ n−1 are again the eigenvalues of √ ∇ T u t ∇ T u, and the signed volume of u is now (1. 6) Let us clarify that here we are using the identification u, (1.7) The constant in (1.5) depends in principle now both on the dimension and on the apriori bound in the L 2(n−2) -norm of the gradient. The reason why this particular condition is introduced will be explained in Subsection 3.4. As we also justify by examples in Remark 3.3, the estimate is optimal in this setting, in the sense that the exponents with which the two deficits appear cannot generically be improved.
For the conformal case, due to the scaling invariant nature of the problem, the correct notions for the average conformal deficit and the isoperimetric one can be combined together. The main result when n = 3 in this case is the following.
Theorem 1.4. There exists a constant c 2 > 0 so that for every u ∈ W 1,2 (S 2 ; R 3 ) with V 3 (u) = 0 there exist a Möbius transformation φ of S 2 and λ > 0 such that where D 2 (u) := 1 2 − S 2 |∇ T u| 2 is the Dirichlet energy of u, and V 3 (u) is again its signed volume, as in (1.4).
Of course the question is void when n = 2, since conformality is a trivial notion for maps from S 1 to R 2 .
The use of this combined conformal-isoperimetric deficit is very natural in this framework. Indeed, generalizing to any dimension n ≥ 3 (for n = 3 cf. [26,Theorem 2.4]), for u ∈ W 1,n−1 (S n−1 ; R n ) the following inequalities, sometimes referred to as Wente's isoperimetric inequality for mappings, are known to hold.
where D n−1 (u), P n−1 (u) are the first two integral quantities in the first line of the above inequalities, i.e., the (n − 1)-Dirichlet energy and the generalized area produced by u respectively. The first inequality in (1.9) follows from the arithmetic mean-geometric mean inequality for the eigenvalues of √ ∇ T u t ∇ T u and equality is achieved iff these eigenvalues coincide for H n−1 -a.e. x ∈ S n−1 , or equivalently, iff (∇ T u) t ∇ T u = |∇ T u| 2 n − 1 I x H n−1 -a.e. on S n−1 , i.e., iff u is a generalized conformal map from S n−1 to R n . The second inequality in (1.9) is the functional form of the isoperimetric inequality (cf. [1, inequality (2)]), which can be proven first for smooth maps and can then be extended by density in W 1,n−1 (S n−1 ; R n ). Equality is achieved iff the image of u is another round sphere in the H n−1 -a.e. sense. In the case of a C 1 embedding, the inequality reduces of course to the classical Euclidean isoperimetric inequality for the open bounded set in R n whose boundary is u(S n−1 ).
Based on these simple observations, the combined conformal-isoperimetric deficit considered among maps u ∈ W 1,n−1 (S n−1 ; R n ) for which V n (u) = 0, provides a correct notion of deficit when one seeks stability of the conformal group of S n−1 among maps from S n−1 into R n . Indeed, it is immediate that E n−1 is translation, rotation and scaling invariant, as well as invariant under precompositions with Möbius transformations of S n−1 . Moreover, as we have discussed above, E n−1 is nonnegative and vanishes iff u is a generalized conformal map from S n−1 onto another round sphere, which after translation and scaling can be taken to be S n−1 again. If d ∈ Z would denote the degree of u ∈ W 1,n−1 (S n−1 ; S n−1 ) (following the definitions in [5]), then Since the degree (for maps from S n−1 to itself) takes integer values, we would have that either d = 0 or d = ±1, with the first case being excluded automatically, since by assumption V n (u) = 0. Hence, absolute minimizers of E n−1 are degree ±1 conformal maps from S n−1 into itself, up to a translation vector and a scaling factor, i.e., according to Theorem 1.1, Möbius transformations of S n−1 up to translation and scaling.
In this respect, Theorem 1.4 can be thought of as a sharp quantitative version of the previous statements for n = 3. At the core of its proof lies the study of the linearized version of the problem, since by the use of a contradiction\compactness argument it is enough to show the theorem for maps that are sufficiently close to the id S 2 in the W 1,2 -topology. In this regime, and after a correct rescaling of u, if w := u − id S 2 is the corresponding displacement field, one obtains the formal Taylor expansion where Q 3 (w) is the associated quadratic form, i.e., the second derivative of E 2 at the id S 2 , defined explicitely later in (4.7). The next and main step of the proof is to examine the coercivity properties of the quadratic form Q 3 . This is something that can actually be done in every dimension n ≥ 3, the main ingredient for doing so being the fine interplay between the Fourier decomposition of a W 1,2 (S n−1 ; R n )-vector field into R n -valued spherical harmonics and the properties of the linear first order differential operator associated to the second derivative of V n at the id S n−1 .
To be more precise, as we thoroughly examine in Subsection 4.2 for the case n = 3, and in Subsection 5.1 for the higher dimensional case, if one rescales u properly, sets w := u − id S n−1 and expands E n−1 (u) in (1.10) around the id S n−1 , then the resulting quadratic form x j ∇ T w j (1.12) has finite-dimensional kernel and its dimension actually coincides with that of the conformal group of S n−1 .
Moreover, when considered in the correct space (see the definitions of the spaces H n , (H n,k,i ) k≥1,i=1,2,3 in equation (4.15) and Theorem 4.7 in Subsection 4.2), the form Q n satisfies the following coercivity estimate.
Theorem 1.5. Let n ≥ 3. There exists a constant C n > 0 such that for every w ∈ H n , 13) where H n,0 := H n,1,2 ⊕ H n,2,3 is the kernel of Q n in H n , and Π n,0 : H n → H n,0 is the W 1,2 -orthogonal projection of H n onto H n,0 .
When n = 3, the optimal constant in (1.13) can actually be calculated explicitely. Since H n,0 turns out to be isomorphic to the Lie algebra of infinitesimal Möbius transformations of S n−1 , an application of the It is maybe worth remarking here that, in contrast to (1.11), in dimensions n ≥ 4 a formal expansion of the combined conformal-isoperimetric deficit around the id S n−1 yields (1.14) Since the higher order term is now cubic in ∇ T w, the linear estimate (1.13) alone would only imply the nonlinear one (following exactly the same steps of proof as those described in Subsections 4.1, 4.3 and 4.4 for the case n = 3) only in the W 1,∞ -close to the id S n−1 -regime (see Remark 5.5), as stated in the following.
Corollary 1.6. Let n ≥ 4. There exist constants θ ∈ (0, 1) (sufficiently small) and c n−1 > 0 such that the following statement holds. For every u ∈ W 1,∞ (S n−1 ; R n ) with ∇ T u − P T L ∞ (S n−1 ) ≤ θ 1, there exist a Möbius transformation φ of S n−1 and λ > 0 such that is the quadratic form associated to the nonlinear conformal deficit D n−1 (u) is the one associated to the nonlinear isoperimetric deficit Actually, an estimate like (1.13) holds true for every positive combination of the two forms Q n,conf and Q n,isop .
Finally, as we mention in Subsection 5.3, a similar linear stability phenomenon holds true in the isometric case as well, namely one can prove the following. Theorem 1.8. Let n ≥ 2. For every α > 0 there exists a constant C n,α > 0 such that for every map where, 16), and w h : B 1 → R n denotes the (componentwise) harmonic continuation of w in the interior of B 1 .
The structure of the paper is the following. In Section 2 we introduce some notations that we are going to use in the subsequent sections. In Section 3 we give in steps the proof of Theorem 1.2 and remark on the adaptations needed to prove its generalization in higher dimensions, i.e., Theorem 1.3. In Section 4 we give again in steps the proof of Theorem 1.4. Building upon the analysis that we perform in Subsection 4.2, in Section 5 we prove the linear stability estimates stated in Theorems 1.5 and 1.8 in all dimensions.
In Appendix A we first exhibit a short, intrinsic and to our knowledge, new proof of Liouville's Theorem 1.1, as well as a related compactness result that can be proven by a slight perturbation of the idea. In Appendix B we include just for the convenience of the reader a detailed derivation of some integral identities for Jacobians, as well as the Taylor expansions of the geometric quantities that appear in the main body of the paper. Finally, in Appendix C we collect some basic facts from the theory of spherical harmonics that we are using.

Notation
The following standard notation will be adopted throughout the paper.
, ·, · , | · | the Euclidean orthonormal basis, inner product, norm in R n A t the transpose of a matrix or the adjoint of the corresponding linear map Sym(n), Skew(n) the space of n × n symmetric, skew-symmetric matrices respectively A sym , A skew the symmetric, skew-symmetric part of a matrix A ∈ R n×n respectively {τ 1 , . . . , τ n−1 } a positively oriented local orthonormal frame for T x S n−1 , so that for every x ∈ S n−1 so that (having in mind the Nash-Kuiper Theorem, cf. [16], [20]) the deficit δ(u) is sharper than the full since it only penalizes local stretches under u. The isoperimetric deficit (or the positive part of the excess in volume) in this setting is denoted by Before presenting the proof of the result, let us make some preliminary remarks.
is a globally short map, then δ(u) = 0. Moreover, since in this case e. on S 2 , by the Cauchy-Schwarz inequality and the sharp Poincare inequality on S 2 (equality in which is achieved for restrictions on S 2 of affine maps of R 3 , see (C.5) in Appendix C), |. This is something that could also be seen just by using the isoperimetric inequality in this case. Hence, for globally short maps only the excess in volume is present in the right hand side of the stability estimate (1.3).
(ii) On the other hand, if u ∈ W 1,2 (S 2 ; R 3 ) is volume-increasing in the sense that |V 3 (u)| ≥ 1, then ε(u) = 0, and only the isometric deficit δ(u) is present in the right hand side of (1.3).
(iii) In all other cases, i.e., if u ∈ W 1,2 (S 2 ; R 3 ) is not globally short and not volume-increasing, both deficits are present in the estimate. It is also immediate that one cannot have simultaneously a globally short map u that is volume-increasing, unless u is a rigid motion of S 2 , something that can be directly verified by checking the equality cases in (3.5).
As we also mentioned in the Introduction, (1.3) is optimal in the norm appearing on the left hand side and the deficits on the right hand side, i.e., the exponent 1 with which δ(u) and ε(u) appear in the estimate cannot generically be improved. Examples showing the optimality of the exponents can easily be constructed even in dimension n = 2, where the exact analogue of Theorem 1.2 becomes There exists a constant c 0 > 0 so that for every u ∈ W 1,2 (S 1 ; Here, ∂ τ u denotes the tangential derivative of u along S 1 . The previous proposition can be proven in exactly the same way as Theorem 1.2, following the arguments of the next subsections. As the reader might observe later, the Lipschitz truncation argument of Subsection 3.1 is even simpler in the case n = 2, because the signed volume V 2 (u) := − S 1 u, (∂ τ u) ⊥ is of first order in ∂ τ u. Keeping the notation δ(u) and ε(u) for the isometric and the isoperimetric deficit also when n = 2, two instructive examples for the optimality of the exponents are given in the next remark.
For each σ ∈ [0, 2π) the map u σ is isometric, being essentially the identity transformation, except for a small circular arc of angle σ, where it is a flip with respect to the horizontal line at height y 0 = sin 3π 2 − σ 2 . Hence, δ(u σ ) = 0 for every σ ∈ [0, 2π). Obviously, ∂ τ u σ → ∂ τ id S 1 strongly in L 2 (S 1 ; R 2 ) as σ → 0 + , and one can easily obtain that On the other hand, using elementary plane-geometry formulas for the area of circular triangles, we can compute the area of the double arc-region of the unit disc missed by u σ , so that also which reveals the optimality of the exponent of ε(u) in the estimate (3.6).
(ii) Identify now S 1 with the interval [0, 1] by identifying the endpoints. For 0 < σ 1, consider the maps and let u σ : S 1 → S 1 be the corresponding maps defined on the unit circle. Obviously, ε(u σ ) = 0 for every σ ∈ [0, 2π). Geometrically, the maps u σ travel back and forth, and produce a triple cover of a small σ-arc, locally stretching S 1 . With similar calculations as before, Moreover, which reveals the optimality of the exponent of δ(u) in the estimate (3.6) in the generic setting. The geometric reason behind this, is the fact that the deficit δ(u) (as well as the full L 2 -isometric deficit δ isom (u)) does not penalize changes in the orientation neither extrinsically, i.e., flips in ambient space, nor intrinsically, when u is seen as a map from the sphere onto its image.
When n = 3 (and also in higher dimensions) one can construct similar examples as in (3.7), (3.8). For instance, in the first case one can consider maps that are the identity outside a small geodesic ball of S n−1 and inside being again flips in R n with respect to the appropriate affine hyperplane. In the second case, one can rotate the previous one-dimensional example around a fixed axis.
We are now ready to present the proof of Theorem 1.2 in steps. For the most part, by straightforward modifications that mainly regard the change of some dimensional constants in the estimates and of some purely algebraic expressions, the arguments are valid in all dimensions, and can be used to prove Theorem 1.3 as well. We will come back to that issue in Subsection 3.4.

Reduction to Lipschitz mappings
As in the pioneering geometric rigidity result of G. Friesecke, R.D. James and S. Müller (cf. [11, Theorem 3.1]), the first step is to justify why it suffices to work with maps with a universal upper bound on their Lipschitz constant. This is achieved through the use of the following standard truncation lemma.
Lemma 3.4. There exists c > 0 so that for every u ∈ W 1,2 (S 2 ; R 3 ) and every M > 0, there exists The proof of this lemma can be performed as for the corresponding statement in the bulk (cf. [ Proof. If δ(u) > 1, recalling the definitions of the deficits in (3.2)-(3.4), we trivially have so we may assume without loss of generality that 0 ≤ δ(u) ≤ 1. With the notation we have employed in (3.1), |∇ T u| 2 = σ 2 1 + σ 2 2 ≤ 2σ 2 2 , and therefore in this case we also have the upper bound (3.10) By a standard argument, using Lemma 3.4, we also obtain (3.11) Indeed, in the set {x ∈ S 2 : |∇ T u(x)| > M := 2 √ 2} we have σ 2 ≥ 1 √ 2 |∇ T u| > 2, so in this set we can estimate pointwise, and then the final estimates in (3.11) follow immediately. (3.12) For the second desired estimate in (3.9) we observe that if V 3 (u M ) > 1 then ε(u M ) = 0 ≤ ε(u), so we may assume without loss of generality that V 3 (u M ) ≤ 1. Then, i.e., it suffices to control the absolute value of the difference between the corresponding signed volumes.
Towards this end, denoting by one can easily verify that where (3.16) We can now estimate each term on the right hand side of (3.15) separately. For the first one, by the isoperimetric inequality (see (1.9) for n = 3) and the second estimate in (3.11), we obtain To estimate the terms (R i (u, u M )) i=1,...,4 we can now use the properties of the Lipschitz truncation u M provided by Lemma 3.4, the Cauchy-Schwarz inequality and the sharp Poincare inequality on S 2 (see (C. 5) in Appendix C), as well as the estimates (3.10) and (3.11), in order to estimate each of the remaining terms in (3.15) as follows.  Having reduced our attention to maps that enjoy an apriori Lipschitz bound, we show in this subsection that for our purposes, we can further assume without loss of generality that the maps in consideration are sufficiently close to the id S 2 in the W 1,2 -topology. To do so, we first prove a qualitative analogue of Theorem 1.2. Recalling the notations introduced in (3.1), (3.4) and (3.14), we have.
Then, there exists O ∈ O(3) so that up to a non-relabeled subsequence, Proof. We can obviously assume without loss of generality that u k := − S 2 u k = 0 for all k ∈ N. Hence, the sequence (u k ) k∈N is uniformly bounded in W 1,2 (S 2 ; R 3 ), and up to passing to a non-relabeled subsequence, converges weakly in W 1,2 (S 2 ; R 3 ) and also pointwise H 2 -a.e. to a map u ∈ W 1,2 (S 2 ; R 3 ) with u = 0. By lower semicontinuity of the Dirichlet energy under weak W 1,2 -convergence, we further have that The last inequality in (3.22) is justified by the following estimates. (3.23) In a similar manner, we can use again the assumption that sup k∈N ∇ T u k L ∞ ≤ cM , and the fact that the determinant is a Lipschitz function, to estimate also and u k = 0, by the sharp Poincare inequality on S 2 (see again (C.5) in Appendix C) and the estimates (3.23), (3.24), we obtain (3.25) By the assumption (3.20), and since u k → u strongly in L 2 (S 2 ; R 3 ), we can let k → ∞ in (3.25), to obtain − S 2 Hence, the limiting map u is such that − S 2 u = 0, and by (3.22) and (3.26) it also satisfies By the equality case in the sharp Poincare inequality on S 2 (since the first nontrivial eigenfunctions of −∆ S 2 are the coordinate functions, cf. Appendix C), we deduce from (3.27) that u(x) = Ax for some A ∈ R 3×3 with |A| 2 = 3. In particular, equalities are achieved in (3.22), and therefore u k → u := Aid S 2 in the strong W 1,2 -topology.
To show that A ∈ O(3), we argue as follows. Having established the strong W 1,2 -convergence of (u k ) k∈N towards u, up to a further non-relabeled subsequence we can assume now that ∇ T u k → ∇ T u also pointwise H 2 -a.e. on S 2 and therefore, using the assumption that ∇ T u k L ∞ ≤ cM for every k ∈ N, (3.28) Using a variant of Lebesgue's Dominated Convergence Theorem in the assumption that lim k→∞ ε(u k ) = 0 and (3.28), allows us to conclude. Indeed, i.e., |detA| ≥ 1. If we now perform the polar decomposition by the arithmetic mean-geometric mean inequality we get and equality in this algebraic inequality implies that α 1 = α 2 = α 3 = 1, i.e., O := A ∈ O(3).
As an immediate consequence of the Lemmata 3.5 and 3.6, we obtain the following.
where c > 0 is the constant of Lemma 3.4, M := 2 √ 2 and 0 < θ 1 is a sufficiently small constant that will be suitably chosen later.
Proof. The proof is a standard contradiction argument. Indeed, suppose that we have proven Theorem 1.2 for maps in A M,θ for some θ ∈ (0, 1) sufficiently small. According to the Lipschitz truncation argument provided by Lemma 3.5, for the general case it suffices to prove that whenever the denominator above is non-zero. Arguing by contradiction, suppose that the latter is false.
Then, for every k ∈ N there exist u k with mean value 0, Lipschitz norm bounded by cM , δ(u k ) + ε(u k ) > 0, In particular, and letting k → ∞ we see that along this sequence, lim k→∞ (δ(u k ) + ε(u k )) = 0. By Lemma 3.6 and up to passing to a subsequence, we can find O 0 ∈ O(3) so that u k → O 0 id S 2 strongly in W 1,2 (S 2 ; R 3 ). Without loss of generality (up to considering O t 0 u k instead of u k if necessary) we can also suppose that O 0 = I 3 , so there exists k 0 := k 0 (θ) ∈ N such that i.e., u k ∈ A M,θ for all k ≥ k 0 . Therefore, by assumption, there should exist (R k ) k≥k 0 ⊂ O(3) (and actually which contradicts (3.30).

Proof of the local version of Theorem 1.2
By the reductions we have performed in the previous two subsections, we are left with proving a local version of Theorem 1.2. This will be done by perturbing quantitatively the idea of proof of Lemma 3.6.
Proposition 3.8. There exists a constant θ ∈ (0, 1) so that for every u ∈ A M,θ (defined in (3.29)), there Proof. First of all, it obviously suffices to prove (3.31) in the regime where both deficits are sufficiently small, say for some absolute constants δ 0 , ε 0 > 0 which will also be chosen sufficiently small later. By using (3.23) with u instead of u k , we have and therefore (3.25), with u instead of u k , would now give us Since by (C.5) we have 1 2 − S 2 |∇ T u| 2 − − S 2 |u| 2 ≥ 0, we can rearrange the terms and use (3.32) to arrive at the estimate the first inequality in which, is justified as follows. Let u h : B 1 → R 3 be the harmonic continuation of u in the interior of B 1 , being taken componentwise. The quantity in the middle of (3.33) is the deficit of u in the L 2 -Poincare inequality for maps with zero average on S 2 . For every k ∈ N, let H k be the subspace of W 1,2 (S 2 ; R 3 ) consisting of vector fields whose components are all k-th order spherical harmonics (see also Appendix C), so that one has the orthogonal (with respect to the W 1,2 -inner product) Let also Π k be the corresponding orthogonal projection. In our case of consideration, Π 0 u = − S 2 u = 0, and it is straightforward to check that Π 1 u = ∇u h (0)x. Since the first non-trivial eigenvalue of the Laplace-Beltrami operator on S 2 is λ 1 = 2 and the second one is λ 2 = 6 (see (C.2)), by orthogonally decomposing u = Π 1 u + (u − Π 1 u), we have (3.34) Hence, the only thing that is left to be justified in order to prove (3.31), is why in (3.33) the matrix can be replaced by a matrix R ∈ SO(3). In that respect, observe that by the mean-value property of harmonic functions, the basic L 2 -estimate C.6 (whose simple proof is given at the end of Appendix C) applied to the function u − id S 2 , and (3.29), we obtain In particular, if θ ∈ (0, 1) is sufficiently small, (3.36) directly implies that (3), the last inequality in (3.37) (in particular the fact that detA > 0) and (3.36) yield the inequality (3.38) can be rewritten as The key observation now is that when θ ∈ (0, 1) is sufficiently small, a map u ∈ A M,θ satisfies the estimate The proof of (3.41) is a bit more involved, and is therefore presented separately in Lemma 3.9. Let us assume for the moment its validity, and see how to finish the proof of (3.31). With the notations introduced in (3.35) and (3.39), we can write expand the polynomial in the eigenvalues and use (3.40), to obtain , after rearranging terms in (3.42) and using (3.41), we get In order to handle the term λ + λ 2 2 , we proceed as follows. Using again the mean value property of harmonic functions, (C.6) applied to u now, and the outcome of (3.23) with u instead of u k here, we can With the notations introduced in (3.35) and (3.39) we have and the last identity, together with (3.44), implies that Since λ does not necessarily have a sign, we distinguish two cases: (i) In the case λ ≤ 0, and since by (3.40) |λ| ≤ √ 3Λ ≤ 3 √ 2 θ 1, the term in the first parenthesis on the right hand side of (3.43) is estimated by since by choosing θ ∈ (0, 1) even smaller if necessary, we can also achieve 1 − 3θ 2 ) is therefore nonpositive in this case, and (3.43) gives (ii) In the case λ > 0, by (3.45) we have 0 < λ ≤ cδ(u), so again (3.43) together with (3.32) imply that In both cases, we obtain and combining (3.46) with (3.33) allows us to deduce (3.31) with R := R 0 ∈ SO(3), and conclude.
To complete the arguments, we finally give the proof of the estimate (3.41), which for convenience of the reader we recall in the next lemma.
Proof. The main trick is to write the signed-volume in the isoperimetric deficit ε(u) as the corresponding bulk integral in B 1 . In particular, using the identity (B.1) (which we prove in Appendix B), we have By the fact that u ∈ A M,θ and (3.37), the map w also satisfies and we will not distinguish further between the universal constants c andc. Because of (3.33), (3.35) and (3.37), we actually get Now, in the rightmost hand side of (3.48) we can use the expansion of the determinant around I 3 , i.e., the identity (B.2) which is proved in Appendix B, according to which, where the quadratic form Q V 3 (w) is explicitly given in (B.3). Notice that the linear term is vanishing, because the definitions of A := ∇u h (0) and w in (3.49), together with the mean value property of harmonic functions, imply that For the higher order terms one can argue as follows. Recalling the notation Π k for the projections onto the subspaces H k of the k-th order spherical harmonics (see the comments after (3.33)), we note that (3.49) and (3.50) directly imply that Π 0 w = Π 1 w = 0. Hence, by the Cauchy-Schwarz inequality and the sharp Poincare inequality on S 2 for w (see the comment just below (C.5)), we obtain and by Wente's isoperimetric inequality (see (1.9) for n = 3), (3.55) Therefore, by (3.54), (3.55) and (3.51), together with the assumption (3.32), we estimate (3.56) In particular, since detA > 0 (see (3.37)), by (3.48), the expansion in (3.52), (3.53) and (3.56), we deduce that V 3 (u) > 0, and we can finally consider two cases: (ii) If 0 ≤ V 3 (u) ≤ 1, then we can again similarly estimate, This finishes the proof of (3.47) in both cases, and allows us to conclude.

The generalization to dimensions n ≥ 4: Proof of Theorem 1.3
By following closely the steps of proof of Theorem 1.2, one can also prove its generalization in dimensions n ≥ 4, i.e., Theorem 1.3. Regarding the extra assumption on an apriori bound in the L 2(n−2) -norm of ∇ T u in the latter, let us first make the following short remark. When n = 3, n − 1 = 2(n − 2) = 2 and the assumption that ∇ T u is apriori bounded in L 2 is obsolete in this case, since we have anyway seen that it suffices to prove Theorem 1.2 for maps u ∈ W 1,2 (S 2 ; R 3 ) for which 0 < δ(u) 1, which trivially implies the bound ∇ T u L 2 ≤ √ 10 (recall (3.10)). In higher dimensions, the assumption is imposed by the growth behaviour of the signed-volume term with respect to ∇ T u.
Indeed, as we will see next, apart from the obvious differences in the proof due to the change in dimension, the only essential difference appears when we are trying to implement the Lipschitz truncation argument of Subsection 3.1, in order to control both the isometric and the isoperimetric deficit of the Lipschitz truncated map in terms of the ones of the original map u.
Proof of Theorem 1.3. Let n ≥ 4, M > 0 and u ∈Ẇ 1,2(n−2) (S n−1 ; R n ) with ∇ T u L 2(n−2) ≤ M . Applying the analogue of Lemma 3.4 in W 1,2 (S n−1 ; R n ) for M n := 2 √ n − 1, we obtain again u Mn ∈ W 1,∞ (S n−1 ; R n ) with ∇ T u Mn L ∞ M n and for which, exactly as in the estimates (3.12) (with σ Mn,n−1 in the place of σ M,2 now) and (3.13) of Lemma 3.5, Since V n is now of order n − 1 > 2 in ∇ T u, it is of course not expected that one can have an estimate of the form of (3.19) without any further assumption, since V n (u) is not even finite if u does not belong to W 1,n−1 (S n−1 ; R n ) at least. Nevertheless, under the imposed assumption that ∇ T u L 2(n−2) ≤ M , the difference of the corresponding signed volumes in (3.57) can be controlled as follows. Assuming again without loss of generality that 0 < δ(u) ≤ 1, adopting the notation in (3.14) and using the fact that By the Sobolev embedding and our assumption, we further have Therefore, by the fact that {u Mn = u} ⊆ {∇ T u Mn = ∇ T u} H n−1 -a.e., (3.60), the first inequality in (3.11) (in dimension n ≥ 4), and the assumption that ∇ T u L 2(n−2) ≤ M , the remainder term in (3.59) can be estimated further by Therefore, under this extra assumption for n ≥ 4, in view of (3.58) and (3.61), (3.57) implies again that Hence, under the assumption that ∇ T u L 2(n−2) ≤ M when n ≥ 4, we can again reduce to proving Theorem For instance, and just for the sake of clarity, we note that in this higher dimensional setting, 1 2 − S 2 |∇ T u| 2 should be replaced in the corresponding arguments by 1 n−1 − S n−1 |∇ T u| 2 , but the arguments go through exactly in the same way, since n − 1 is both the norm of the gradient of isometric maps from S n−1 to R n and also the first nontrivial eigenvalue of −∆ S n−1 (the second being 2n, see Appendix C and (3.34)). In this respect, the estimate (3.33) should of course be replaced by and analogously to (3.52), the expansion of the signed-volume around the identity is now Modulo these changes, the proof remains essentially unchanged, which is left to the reader to verify.
10. An interesting question would be whether for n ≥ 4 the apriori bound on the L 2(n−2)norm of ∇ T u, imposed as an assumption in Theorem 1.3, can be replaced by one in an L p -norm for some p ∈ (n − 1, 2(n − 2)). The previous approach indicates however that the exponent 2(n − 2) = (n − 1) + (n − 3) in the assumption is the sharpest one.
Indeed, let us assume that ∇ T u L n−1+γ ≤ M for some γ ∈ (0, n−3) and M > 0. Then, for α ∈ (0, n−1) and p > 1, we can apply Hölder's inequality and use again the analogues of the estimates in (3.11) for n ≥ 4, to deduce as before that As long as αp ≤ 2, by Hölder's inequality again, and (3.11) (for n ≥ 4) would finally give us as long as 1 ≤ p(n−1−α) p−1 ≤ n − 1 + γ. Therefore, by (3.63) and following closely the estimates used to arrive at (3.58) and (3.61), we deduce that the optimal exponent with which δ(u) can appear in (3.62) through these estimates is exactly But this value should also satisfy the inequality αp ≤ 2, which implies that where D 2 (u), V 3 (u) are as in the statement of Theorem 1.4, and that E 2 (u) = 0 iff u is a Möbius transformation of S 2 , up to a translation vector and a dilation factor.
To pass from the nonlinear deficit E 2 to its linearized version, we make use of the following compactness result, whose proof can be found for instance in [4] or [6, Lemma 2.1] (stated on R 2 rather than S 2 therein).
Using this compactness lemma and the invariances of the combined conformal-isoperimetric deficit, with a contradiction argument as the one we used in the proof of Corollary 3.7, one can now prove the following.
Proof. The fact that without loss of generality we can assume (i) is obvious because (1.8) is translation invariant. That we can assume property (iii) is also immediate, because for every u ∈ W 1,2 (S 2 ; R 3 ) with it suffices to prove the desired estimate in the small-deficit regime.
Suppose now that we have proven Theorem 1.4 for maps in B θ,ε 0 , but for the sake of contradiction the theorem fails to hold globally. Then, for every k ∈ N there exists We can then use Lemma 4.1, and argue as in the end of the proof of Corollary 3.7, to arrive at a contradiction for the corresponding maps Having now reduced to showing Theorem 1.4 for mappings in B θ,ε 0 , where V 3 (u) > 0, we can linearize the initial problem, by making use of the following two lemmata.
Lemma 4.5. There exists a constant β := β(θ, ε 0 ) > 0 that tends to 0 as (θ, ε 0 ) → (0, 0), such that the following holds. If u ∈ B θ,ε 0 satisfies (4.3) and one sets w := u − id S 2 , then Proof. For u as in the statement of the lemma, property (4.3) can be rewritten as Then, (4.9) Since d 2 (1 + t) 3 2 = 3 4 , we can take θ ∈ (0, 1) small enough so that (by (4.2)(ii)) the higher-order term in the expansion (4.9) is estimated by Regarding the expansion of the signed volume term V 3 (u), as we calculate in detail in Lemma B.1 and by using (4.8), we have Hence, by using the expansions (4.9) and (4.11) in the definition (4.1) of the deficit, we obtain and after rearranging terms, Arguing exactly as in (3.54) (with the Poincare inequality being applied with constant 1 2 instead of 1 6 in this case) and (3.55), and using again (4.2)(ii), we have Therefore, (4.13), the estimates (4.10) and (4.14) for the remainder terms, and (4.2)(iii) imply that where the precise value of the constant is β : In view of Lemma 4.5, if we thus choose ε 0 ∈ (0, 1) sufficiently small and then θ ∈ (0, 1) sufficiently small accordingly, the last term on the right hand side of (4.6) can be set to be a sufficiently small multiple of the Dirichlet energy of w. Therefore, we can move our focus of attention on the coercivity properties of the resulting quadratic form Q 3 defined in (4.7), which is just the second derivative of the nonlinear combined conformal-isoperimetric deficit E 2 (u) at the id S 2 . This will be the content of the next subsection.

On the coercivity of the quadratic form Q 3
For the most part of this subsection the results hold true in every dimension n ≥ 3. Since we will use them also in Section 5, where we prove linear stability estimates in all dimensions, we also denote here the ambient dimension 3 with the general letter n (in order to avoid the repetition of the arguments in Section 5), and hope that no confusion will be caused to the reader. Our goal is to examine the coercivity properties of the quadratic form Q n in (4.7). By the reductions we have performed (see (4.2) and (4.3)), this can be considered in the space Similarly to the notation introduced in the proof of Proposition 3.8 in Subsection 3.3, for every k ≥ 1 we define H n,k to be the linear subspace of H n consisting of those maps in H n , all the components of which are k-th order spherical harmonics (cf. also Appendix C), and also definẽ so that ∞ k=1H n,k is a W 1,2 -orthogonal decomposition of the vector space of harmonic maps w h : B 1 → R n for which w h (0) = 0 and Tr∇w h (0) = 0, the last identities following immediately from their equivalent formulation on S n−1 in (4.15). Actually, for every k ≥ 1 we can further consider the W 1,2 (B 1 )-Helmholtz decompositionH n,k =H n,k,sol ⊕H ⊥ n,k,sol , andH ⊥ n,k,sol is its orthogonal complement in W 1,2 (B 1 ; R n ). In view of the k-homogeneity of the maps iñ H n,k in (4.16), we can write the equivalent to (4.17) W 1,2 -decomposition also on S n−1 , namely where H n,k,sol := w ∈ H n,k : w h ∈H n,k,sol , (4.20) and H ⊥ n,k,sol is its W 1,2 (S n−1 ; R n )-orthogonal complement. Hence, adopting from now on all these notations introduced in (4.15)-(4.20), let us also denote N n,k := dimH n,k < ∞, N 1,n,k := dimH n,k,sol , N 2,n,k := dimH ⊥ n,k,sol , (4.21) so that N n,k = N 1,n,k + N 2,n,k .
Recall also that the second derivative of the signed-volume term V n at the id S n−1 corresponds to the bilinear form where the associated linear first-order differential operator A is defined as x j ∇ T w j for w ∈ H n . The main feature that we are going to use in this subsection is the fine interplay between the operator A and the above defined spaces, as it is properly described in the following. Proof. First of all, it is immediate that A is self-adjoint with respect to the L 2 -inner product in H n , since it arises as the second derivative of V n at the id S n−1 , but it is also easy to verify directly after integrating by parts that for any v, w ∈ H n , Note also that, since (H n,k,sol ) k≥1 and (H ⊥ n,k,sol ) k≥1 are subspaces of the k-th order spherical harmonics, the W 1,2 -and the L 2 -inner products restricted on these subspaces are equivalent (see (C.2)). It is also easy to check that for every k ≥ 1, w ∈ H n,k =⇒ A(w) ∈ H n,k . (4.24) Indeed, as we mention in the beginning of Appendix C, for k ≥ 1 fixed and w ∈ H n,k , its harmonic extension w h in B 1 is an R n -valued homogeneous harmonic polynomial of degree k, and ∀j = 1, 2, . . . , n, Therefore, the operator A can alternatively be rewritten as x j ∇w j h on S n−1 . (4.25) Writing A in terms of the full gradient and divergence operators on S n−1 as in (4.25), we see that and by (4.15), It is also straightforward to verify that so [A(w)] h is also an R n -valued homogeneous harmonic polynomial of degree k, and therefore its restriction on S n−1 is an R n -valued k-th order spherical harmonic. In total, (4.26)-(4.28) yield the implication in (4.24).
Directly from (4.28) one can also verify that A leaves H n,k,sol invariant, i.e., as well. It remains to be checked that Indeed, let w ∈ H n,k,sol be such that Note that since w ∈ H n,k,sol is the restriction on S n−1 of an R n -valued homogeneous harmonic polynomial of degree k (hence smooth up to the boundary), the above equation holds true in the classical sense. Hence, by orthogonality, both the normal and the tangential part of A(w) would have to vanish identically, namely, div S n−1 w = 0 and n j=1 x j ∇ T w j = 0 on S n−1 . (4.30) By the definition of H n,k,sol in (4.20) we have that divw h ≡ 0 in B 1 , and therefore (4.30) implies that Testing now the second one of the equations in (4.30) with the vector field w itself, integrating by parts on S n−1 and using (4.31), we obtain i.e., w ≡ 0 on S n−1 . This concludes the proof of (4.29), and thus the proof of the fact that A is a selfadjoint linear isomorphism of H n,k,sol . Hence, A leaves H ⊥ n,k,sol invariant as well, and is actually also an isomorphism of it, as we will see next.  where H n,k,1 is the eigenspace of A corresponding to the eigenvalue σ n,k,1 := −k and H n,k,2 is the one corresponding to the eigenvalue σ n,k,2 := 1.
(ii) For every k ≥ 1, the subspace H n,k,3 := H ⊥ n,k,sol is an eigenspace with respect to A corresponding to the eigenvalue σ n,k,3 := k + n − 2 .
Proof. As we have just remarked before the statement of Theorem 4.7, for every k ≥ 1 there exists a W 1,2 -orthonormal basis of eigenfunctions {w n,k,1 , . . . , w n,k,N 1,n,k } for the subspace H n,k,sol (see also (4.21)) and similarly, {w n,k,N 1,n,k +1 , . . . , w n,k,N n,k } for H ⊥ n,k,sol , i.e., for i = 1, . . . , N n,k , the map w n,k,i satisfies the eigenvalue equation For each such eigenvalue σ n,k,i we denote its corresponding eigenspace by H n,k,i . If in (4.34) we take the inner product with the unit normal vector field on S n−1 , we obtain further that each eigenfunction w n,k,i satisfies the equation div S n−1 w n,k,i = σ n,k,i w n,k,i , x on S n−1 , (4.35) which in terms of the full divergence can be rewritten as div(w n,k,i ) h = div S n−1 w n,k,i + ∂ ν (w n,k,i ) h , x = (σ n,k,i + k) w n,k,i , x on S n−1 . (4.36) We now fix the index k ≥ 1 and consider all the different possible cases that will allow us to find the eigenvalues of A in the invariant subspaces H n,k,sol and H ⊥ n,k,sol respectively.
(a 1 ) Let w be a non-trivial eigenfunction of A in H n,k,sol , so due to (4.16), (4.18) and the (k − 1)-homogeneity of divw h in this case. By (4.36) we see that one possibility for (4.37) to hold, is for the eigenvalue σ = −k. We thus set σ n,k,1 := −k and label its corresponding eigenspace as H n,k,1 := span{w n,k,1 , . . . , w n,k,p n,k } , where p n,k := dimH n,k,1 .
(a 2 ) Let now w be a non-trivial eigenfunction of A in H n,k,sol , with w ∈ H ⊥ n,k,1 . Then, in view of (4.36), the only possibility for (4.37) to hold is iff w, x ≡ 0 on S n−1 . x j ∇ T w j on S n−1 .
(b) Let us now look at eigenfunctions w of A in H ⊥ n,k,sol , where the divergence of w h ∈H n,k in (4.16) does not vanish identically in B 1 . Since w h is an R n -valued k-homogeneous harmonic polynomial, we have that divw h is a scalar (k − 1)-homogeneous harmonic polynomial, and therefore its restriction on S n−1 is a scalar (k − 1)-spherical harmonic. We can then apply the Laplace-Beltrami operator (see (C.1)) on both sides of (4.36) and use again (4.35), to obtain Since in this case divw h does not vanish identically, we conclude that We label this eigenvalue as σ n,k,3 := k+n−2 and its corresponding eigenspace as H n,k,3 . In particular, we have found that H ⊥ n,k,sol = H n,k,3 .  The triviality of H n,1,3 is a consequence of the fact that we had already scaled properly our initial maps u, so that the corresponding maps w satisfy − S n−1 w, x = 0 (recall (4.15)). Indeed, let w(x) := Λx ∈ H n,1,3 for some Λ ∈ R n×n . By assumption, Therefore, divw h ≡ TrΛ ≡ 0 in B 1 , i.e., w ∈ H n,1,sol = H ⊥ n,1,3 , forcing w ≡ 0, and thus, H n,1,3 = {0}.
The eigenvalue decomposition (4.40) of H n into eigenspaces of A is valid for every n ≥ 3. In the case of interest of this subsection, i.e., in dimension n = 3, it immediately gives the desired coercivity estimate for the quadratic form Q 3 defined in (4.7), with optimal constant. For the rest of this subsection we switch back to denoting the ambient dimension by the number 3. As a consequence of Theorem 4.7, we have Lemma 4.9. The following statements hold true.
In particular, for i = 1, 2, 3, if w ∈ H 3,k,i , by the definition (4.22) (for n = 3) and Theorem 4.7, we have which is precisely the first identity in (4.41) for c 3,k,i := 3σ 3,k,i 2λ 3,k , and then where C 3,k,i := 3 4 − c 3,k,i . We list below the precise values of the constants, which are important in this case, since we will need to sum up the identities for Q 3 in the subspaces (H 3,k,i ) k≥1,i=1,2,3 , in order to obtain an estimate on the full space H 3 .
As an immediate consequence of Lemma 4.9 we obtain the desired estimate for the quadratic form Q 3 defined in (4.7).

Proof of the local version of Theorem 1.4
The presence of the Q 3 -degenerate space H 3,0 in the coercivity estimate (4.44) is a small but natural obstacle to overcome in order to complete the proof of Theorem 1.4. As we mentioned after the statement of Theorem 1.5 in the Introduction, since H 3,0 will eventually turn out to be isomorphic to the Lie algebra of infinitesimal Möbius transformations of S 2 , at an infinitesimal level this basically means the following.
Although, by the reductions we have made, the map u is apriori supposed to be θ-close to the id S 2 in the W 1,2 -topology (recall (4.2)(ii)), there might be another Möbius transformation of S 2 that is also θ-close to the id S 2 and is a better candidate for the nearest Möbius map to u in terms of its combined conformalisoperimetric deficit E 2 (u) in (4.1). Similarly to [10] and [22], an application of the Inverse Function Theorem and a topological argument will allow us to identify this more suitable candidate, see the details in the subsequent Lemma 4.13 and its proof. For this purpose, we will need the following characterization of the Q 3 -degenerate subspace H 3,0 , which is valid in every dimension n ≥ 3, and that is why we now switch back to denoting the ambient dimension by n.
Lemma 4.11. The following statements hold true.
(i) The subspace H n,1,2 in (4.33) can be characterized as
The projection on H n,2,3 is therefore characterized by Proof. For part (i) of the lemma, if w ∈ H n,1,2 we can write it as w(x) = Λx for some Λ ∈ R n×n . In this space, recalling (4.38), The characterization (4.48) of the projection Π H n,1,2 is then immediate. For part (ii), let w ∈ H n,2,sol . By for each k = 1, . . . , n, there exists Λ k ∈ Sym(n) such that In particular, for each k, l = 1, . . . , n, we can compute Remark 4.12. It is worth noticing here that simply by counting dimensions, which is also the dimension of Conf (S 2 ) (recall the notation in Section 2 and see also Remark A.1 for some more details on this Lie group and its corresponding Lie algebra). We therefore only need to verify that H 3,0 (introduced after (4.44)) is actually isomorphic to the Lie algebra of infinitesimal Möbius transformations of S 2 , and then the Inverse Function Theorem can be applied. This is the context of the following.
Lemma 4.13. Given θ, ε 0 ∈ (0, 1) sufficiently small, there existsθ ∈ (0, 1) that depends only on θ and is sufficiently small as well, so that for every u ∈ B θ,ε 0 (as in (4.2)) there exists φ ∈ Conf + (S 2 ) such that Proof. Given u ∈ B θ,ε 0 , in view of the characterizations (4.48) and (4.50), let us introduce the map (4.53) Our goal for (4.52) is essentially to show that 0 ∈ Im(Ψ u ). To simplify notation, let us also set Ψ := Ψ| id S 2 . Clearly, Ψ(id S 2 ) = 0. In order to apply the Inverse Function Theorem, we look at the differential and prove that it is a non-degenerate linear map. The differential of Ψ at the id S 2 is easy to compute.
Indeed, by the linearity of all the operations involved, for every Y ∈ T id S 2 Conf + (S 2 ), defined via (see (A.11) in Remark A.1 for the derivation of this representation), and with a slight abuse of notation in the domain of definition of Ψ in (4.54), we can calculate

(4.56)
It is clear that the harmonic extension in B 1 of Y as in (4.55) is given by the vector field In particular, (4.57) directly implies that Therefore, in view of (4.56) and (4.58), we indeed obtain ker(dΨ| id S 2 ) = {0}, i.e., dΨ| id S 2 is a linear isomorphism between T id S 2 Conf + (S 2 ) and R 6 .
From now on, we denote by D σ the open ball in the 6-dimensional vector space T id S 2 Conf + (S 2 ), centered at 0 (or better said, at id S 2 ) and of radius σ > 0. Since the exponential mapping exp id S 2 (·) is a local diffeomorphism between a neighbourhood of 0 in T id S 2 Conf + (S 2 ) and a neighbourhood of id S 2 in Conf + (S 2 ), we can use the Inverse Function Theorem to find a sufficiently small σ 0 ∈ (0, 1) such that for the open neighbourhood U 0 := exp id S 2 (D σ 0 ) (4.59) of the id S 2 in Conf + (S 2 ), the map As a next step, we justify that Ψ is homotopic to Ψ u in U 0 . Indeed, for every φ ∈ U 0 , we can estimate (4.61) In the last step of (4.61) we used the general estimate the last inequality following from the second estimate in (C.7), which is proved in Lemma C.2. Note now that by the conformal invariance of the Dirichlet energy in two dimensions and (4.2)(ii), By the change of variables formula, (4.2) again, and the Poincare inequality (C.5) (since − S 2 (u − id S 2 ) = 0), we also have The strict positivity of the constant C 1 (U 0 ) in (4.64) is ensured by the fact that we can take the neighbourhood U 0 to be sufficiently small around id S 2 , which amounts to choosing σ 0 ∈ (0, 1) sufficiently small (see (4.59) and (4.60)). Hence, (4.61)-(4.64) imply that We can now continue as in [10,Proposition 4.7]. For the sake of making the proof self-contained, we present the argument here, adapted to our setting.

Proof of Theorem 1.4
We can now combine all the previous steps to complete the proof of the main theorem for the conformal case in dimension 3.
Proof of Theorem 1.4. In view of Corollary 4.2, for θ ∈ (0, 1) and ε 0 ∈ (0, 1) that will be chosen sufficiently small in the end, let us consider a map u ∈ B θ,ε 0 . By first using Lemma 4.13 and then Lemma 4.4, we can find a Möbius transformation φ ∈ Conf + (S 2 ) such that the map has all the desired properties, i.e., where we have abused notation by not replacing θ withθ. Settingw :=ũ − id S 2 , we can again expand the deficit around the id S 2 and arrive at (4.6). This estimate, together with (4.44) and the fact thatw ∈ H 3 is such that Π 3,0w = 0 (by (4.15) and (4.73)) yield By Lemma 4.5, we can then choose ε 0 ∈ (0, 1) small enough and subsequently θ ∈ (0, 1) small enough, so that β ≤ 1 8 , in order to absorb the last term on the right hand side of (4.74) in the one on the left, and conclude for φ as above and λ := − S 2 u • φ, x > 0 .
5 Linear stability estimates for Isom(S n−1 ) and Conf (S n−1 ) As we have seen in the Section 4, the key step in proving Theorem 1.4 consists in establishing the corresponding linear estimate, i.e., Theorem 4.10. What we would like to present in this section, is how the eigenvalue decomposition of H n into eigenspaces of A, as provided by Theorem 4.7, which is valid in every dimension (actually even in dimension n = 2), can be used to prove the analogous linear estimate for the quadratic form Q n introduced in (1.12), also in dimensions n ≥ 4. As explained in the Introduction, this quadratic form is associated with the combined conformal-isoperimetric deficit E n−1 defined in (1.10) (cf. Appendix B for the precise calculations). We discuss in detail this case first, since the proof of the corresponding estimate for the isometric case, i.e., Theorem 1.8, is essentially the same and is discussed in

Proof of Theorem 1.5
For n = 3, Theorem 1.5 is precisely Theorem 4.10, whose proof was given in Subsection 4.2, and actually the optimal constant for the linear estimate (4.44) was calculated. In the higher dimensional case n ≥ 4, the quadratic form Q n in (1.12) has an extra − S n−1 (div S n−1 w) 2 -term, and the study of its coercivity properties is slightly more complicated than before.
Remark 5.1. An abstract way to obtain the estimate (1.13) of Theorem 1.5 would be to identify the kernel of Q n in H n (see (4.15)) (and prove that it is exactly the subspace H n,0 ) and then use a standard contradiction\compactness argument (that we describe in Subsection 5.3 for the corresponding result in the isometric case). Notice that w ∈ H n lies in the kernel of the nonnegative quadratic form Q n iff Q n (v, w) = 0 ∀v ∈ H n , i.e., iff w ∈ H n lies in the kernel of the associated Euler-Lagrange operator x j ∇ T w j , (5.1) in the sense of distributions. When n = 3, the second term in (5.1) is dropping out, and since L leaves the subspaces (H 3,k,sol ) k≥1 , (H ⊥ 3,k,sol ) k≥1 invariant in this case, the Euler-Lagrange equation can be solved explicitely, showing that kerL = H 3,0 (as Theorem 4.10 describes quantitatively). Although in slightly hidden form, this was essentially the point of Lemma 4.6 and the subsequent results in Subsection 4.2.
In higher dimensions n ≥ 4, since the operator ∇ T div S n−1 w − (n − 1)(div S n−1 w)x neither commutes with A ((4.23)), nor leaves the subspaces (H n,k,sol ) k≥1 , (H ⊥ n,k,sol ) k≥1 invariant, it is not clear if there is a straightforward argument to solve the equation L(w) = 0 explicitely, and show that indeed kerL = H n,0 in this case as well. In particular, (comparing with (4.46)), mixed terms of special type are expected to be present in the Fourier decomposition of Q n into the eigenspaces of A, which were identified in Theorem 4.7. Nevertheless, it will turn out that we can still use the latter to show that the presence of the mixed divergence-terms is harmless, i.e., it does not produce any further zeros (other than H n,0 ) in Q n . Simultaneously, we obtain the desired coercivity estimate (1.13) (with an explicit lower bound for the optimal constant) by examining how Q n behaves in each one of the eigenspaces (H n,k,i ) k≥,i=1,2,3 of A separately.
Following the notation we had in Subsection 4.2, we first present two auxiliary lemmata that entail most of the essential ingredients for the proof of Theorem 1.5 also in dimensions n ≥ 4.
Lemma 5.2. For n ≥ 3 and k ≥ 1, let us denote by λ n,k := k(k + n − 2) the eigenvalues of −∆ S n−1 (see (C.2)) and let i = 1, 2, 3. For every w ∈ H n,k,i (as in Theorem 4.7), we have Proof. The first identity is immediate from (4.22), (4.23), Theorem 4.7 and (C.3). For the second one, after integration by parts we see that the quadratic form Q Vn can be equivalently rewritten as Together with the first identity and (4.35), (5.3) yields the desired identity, and then the one for Q n follows immediately by its definition in (1.12) and the two previous identities that we just checked.
Since they will again play an important role in the sequel (as it was the case for the constants in (4.43)), let us list below the precise values of the previous constants appearing in (5.2). The last set of constants in the following table is considered for k ≥ 2, because in any case Π H n,1,3 w = 0 for every w ∈ H n , as we have justified in Remark 4.8.
For k ≥ 1; The next ingredient we need is the following.
Lemma 5.3. The following statements hold true.
(i) Let n ≥ 3. For every k, l ≥ 1 and i, j = 1, 2, 3 with (k, i) = (l, j), the subspaces H n,k,i and H n,l,j (introduced in Theorem 4.7) are Q Vn -and Q n -orthogonal, where i.e., for every w n,k,i ∈ H n,k,i and w n,l,j ∈ H n,l,j , Q Vn (w n,k,i , w n,l,j ) = 0 and Q n (w n,k,i , w n,l,j ) = 0 . w n,k,i , where w n,k,i ∈ H n,k,i . Then, Remark 5.4. Before giving the proof of Lemma 5.3, let us point out an interesting feature in formula (5.7), which will be useful in the proof of Theorem 1.5, namely that the summation in the last term of the expression starts from k = 3. The reason for this is that in any case w n,1,3 ≡ 0 whenever w ∈ H n (see In order to prove (5.8), recall that by definition (see (4.18), (4.20) and (4.33)), div(w n,4,1 ) h ≡ 0 in B 1 , and by also using (4.35) for (k, i) = (2, 3), (4, 1) and the divergence theorem, we obtain − S n−1 div S n−1 w n,2,3 div S n−1 w n,4,1 = −4n− To justify that the last integral on the right hand side of (5.9) is zero, observe that because − B 1 (w n,4,1 ) h , (w n,2,3 ) h = 0. Moreover, we observe that for every i = 1, . . . , n, Since (w n,2,3 ) h is an R n -valued 2nd-order homogeneous harmonic polynomial, ∂ i div(w n,2,3 ) h is simply a constant. But then, (5.11) implies that the function is a homogeneous harmonic polynomial of degree 2, hence also L 2 -orthogonal to (w n,4,1 ) i h . Thus, where we used the fact that the function (w n,4,1 ) i h |x| 2 is 6-homogeneous, so that we can write its integral over B 1 as an integral over S n−1 , up to the correct multiplicative constant. The last integral in (5.12) is of course zero for every nontrivial spherical harmonic. Therefore, (5.9)-(5.12) imply (5.8). Note that the previous argument relies on the fact that ∂ i div(w n,2,3 ) h is constant, and of course cannot be implemented for the mixed terms of higher order.
Proof of Lemma 5.3. As in Lemma 4.9, part (i) is an immediate consequence of the fact that the subspaces (H n,k,i ) k≥1,i=1,2,3 are mutually orthogonal in W 1,2 (S n−1 ; R n ). For part (ii), having established that the form Q n in (5.5) splits completely in the eigenspaces (H n,k,i ) (k,i)∈N * ×{1,2,3}\ (1,3) , what remains to be checked is that whenever w n,k,i ∈ H n,k,i and w n,l,j ∈ H n,l,j , there holds − S n−1 div S n−1 w n,k,i div S n−1 w n,l,j = 0 This can be checked again using the different equivalent formulas for Q Vn . Since div S n−1 w ≡ 0 whenever w ∈ H n,k,2 (see (4.35) and (4.38)), we may suppose without loss of generality that i, j ∈ {1, 3}, and then by (5.3) (written now in its bilinear expression), Q Vn (w n,k,i , w n,l,j ) = n 2 − S n−1 div S n−1 w n,k,i w n,l,j , x + n 2 − S n−1 div S n−1 w n,l,j w n,k,i , x − n 2 2 − S n−1 w n,k,i , x w n,l,j , x + n 2 − S n−1 w n,k,i , w n,l,j . (5.14) Actually, as we verified in part (i), Q Vn (w n,k,i , w n,l,j ) = 0, and in view of (4.35), (5.14) yields (σ n,k,i + σ n,l,j − n) − S n−1 div S n−1 w n,k,i div S n−1 w n,l,j = 0 , i.e., (5.13) holds, unless the pairs (k, i) = (l, j) are such that σ n,k,i + σ n,l,j = n. In this respect, (iii) If i = 1, j = 3, σ n,k,i + σ n,l,j = n ⇐⇒ −k + l + n − 2 = n ⇐⇒ l = k + 2 , (iv) If i = 3, j = 1, σ n,k,i + σ n,l,j = n ⇐⇒ k + n − 2 − l = n ⇐⇒ k = l + 2 , which proves the desired claim and then the formula (5.7) for Q n follows by the bilinearity of the expression, and the observation we made in Remark 5.4.
We now have all the necessary ingredients to prove Theorem 1.5. As a preliminary remark, let us note that by taking a closer look at the values of the constants (5.4) of Lemma 5.2, we see that in the case n ≥ 4 one cannot merely neglect the − S n−1 (div S n−1 w) 2 -term and argue exactly as in the proof of Theorem 4.10 (n = 3), something that was also indicated in Remark 5.1. Indeed, although the quadratic form Q n (w) in (5.5) is splitting among the eigenspaces (H n,k,i ) k≥1,i=1,2,3 , for n ≥ 4 it does not have a sign. Actually, Q n (w) is negative in H n,k,3 for every k = 2, . . . , n − 2, zero in H n,1,2 , H n,n−1,3 and strictly positive in each one of the other eigenspaces. On the other hand, we see that the quadratic form Q n vanishes again in the space H n,0 := H n,1,2 ⊕ H n,2,3 .
which finishes the proof of (1.13). For n ≥ 4, the constant C n in (5.23) provides an explicit lower bound for the value of the optimal constant for which (1.13) holds. Another well known fact regarding the connection of the quadratic form in the right hand side of (5.24) to the geometry of R + SO(n) is the following (see [23, Chapters 2 and 3], or [10] for more details). If T R + SO(n) stands for the tangent space to the conformal group R + SO(n) at I n , it is immediate that so that the function A → d(A) := A sym − TrA n I n is equivalent to the distance of A from T R + SO(n). Therefore, the linear subspace Σ n := u ∈ W 1,2 (R n ; R n ) : (∇u) sym = divu n I n can be viewed as the Lie algebra of the Möbius group of R n . If Π Σn : W 1,2 (U ; R n ) → Σ n is the W 1,2projection on this finite-dimensional subspace, the following variant of Korn's inequality for the trace-free part of the symmetrized gradient is known to hold. C := C(n, U ) > 0 such that for every w ∈ W 1,2 (U ; R n ), .
In a certain sense, Theorem 1.5 is the analogue of Theorem 5.6 for maps from S n−1 to R n . In particular, as an upshot of it we have encountered the following fact. Although the kernels of the nonnegative quadratic forms arising as the second derivatives of the conformal deficit D n−1 (u) Indeed, by the calculations that we exhibit in Lemma B.3 in Appendix B, we have that and Q n,isop (w) := 1 2 The nonnegative quadratic form Q n,conf in (5.25) corresponds to the one associated to the conformal deficit in (5.24) for maps defined in the bulk, but the kernel of Q n,conf is infinite-dimensional. Intuitively, the underlying geometric reason behind this, is the abundance of C 2 conformal maps from S n−1 into R n .
Actually, we observe that for every φ ∈ W 1,2 (S n−1 ; R) with − S n−1 φ = 0 and − S n−1 φ(x)x = 0, the map w φ (x) := φ(x)x belongs to H n (recall (4.15)) and one can easily verify that Since the space of such φ is infinite-dimensional, we have in particular that dim(ker(Q n,conf )) = ∞.
Regarding the quadratic form Q n,isop in (5.26), we have that it is also nonnegative and its kernel is also infinite-dimensional. In fact (with the notation of Theorem 4.7), one can directly check that To verify (5.27) we use the following identity, referred to as Korn's identity, that is interesting in its own right and whose derivation is a simple calculation which is also included at the end of Appendix B.
Lemma 5.7. (Korn's identity on S n−1 ). For every w ∈ W 1,2 (S n−1 ; R n ) the following identity holds The interesting point of this identity is that when n ≥ 3, the quadratic form Q Vn appears in the right hand side of (5.28) as some short of curvature contribution and it is really a surface identity, in the sense that the corresponding identity in the bulk is but the last term on the right hand side of (5.29) should now be interpreted as a boundary-term contribution.
By using Korn's identity, the form Q n,isop can be rewritten in a simpler form as But if w ∈ H n,2 , then − n j=1 x j ∇ T w j = A(w) = w on S n−1 (recall that (4.34), (4.39) are then satisfied with σ = 1), and therefore Q n,isop (w) = 0, which proves the implication in (5.27).

Proof of Theorem 1.8
As we have mentioned in the Introduction, if u ∈ W 1,2 (S n−1 ; R n ) and w := u − id S n−1 , then the full L 2 -isometric deficit of u is formally expanded around the identity as For n = 2 we have that dim(ker(Q 2,isom )) = ∞. Indeed, for v(x) := ψ(x)τ (x) + φ(x)x : S 1 → R 2 , where φ, ψ ∈ C ∞ (S 1 , R) and τ (x) := (−x 2 , x 1 ) is the unit tangent vector field on S 1 , it is an easy calculation to For n ≥ 3, and unlike Q n,conf , which as we have seen has infinite-dimensional kernel, Q n,isom has finite-dimensional kernel and actually ker(Q n,isom ) Skew(n) so(n) , this fact being known as the infinitesimal rigidity of the sphere. The reader is referred to [24,Chapter 12] for a detailed discussion and further references regarding this well known geometric fact that is the linear analogue of the C 2 -rigidity of the sphere in the Weyl problem.
Remark 5.8. Without referring to the classical proof of the infinitesimal rigidity of S n−1 , it is very easy to deduce in particular that ker(Q n,isom ) ∩ ker(Q n,isop ) so(n) . (5.31) Indeed, if w ∈ W 1,2 (S n−1 ; R n ) lies in the common null-space of these two nonnegative forms, we have (P t T ∇ T w) sym = 0 and Q n,isop (w) = 0 . Remark 5.9. Even though for n ≥ 3 we have ker(Q n,isom ) so(n), an estimate of the type does not hold, the obstacle being (loosely speaking) the derivatives of the normal component of w. For example, if one considers purely normal displacements w φ (x) := φ(x)x with φ ∈ W 1,2 (S n−1 ; R), then by a straightforward computation one can check that whereas the full gradient of w φ also has derivatives of φ in it. In coordinates, so if the estimate above was to be valid, it would resemble some short of reverse-Poincare inequality, which is of course generically false. Combining additively Q n,isom and Q n,isop has though the merit of providing a Korn-type inequality in terms of the full gradient, as described in Theorem 1.8.
Proof of Theorem 1.8. For α > 0 let us call Q n,α (w) := αQ n,isom (w) + Q n,isop (w) . With the notation we had introduced in Subsection 4.2, the estimate to be proven for Q n, n n−1 reads Once we have (5.38), the case of general α > 0 is immediate, since: The proof of (5.38) can now be performed by following exactly the same procedure as in the proof of Recalling the values of the constants from (5.2) and the table (5.4), we have that C n,k,i > C n,k,i for i = 1, 3, while C n,k,2 = C n,k,2 (because α n,k,2 = 0) .
Remark 5.10. Since in this case we had an easy argument to infer that ker(Q n,isom ) ∩ ker(Q n,isop ) so(n) (see Remark 5.8), the proof of Theorem 1.8, for α = n n−1 for example, could also be performed by a standard contradiction\compactness argument, for an abstract constant though. Indeed, suppose that for α := n n−1 the estimate (1.17) (or equivalently (5.38)) is false. By translation and scaling invariance of the estimate, there exists a sequence (w k ) k∈N ⊂ W 1,2 (S n−1 ; R n ) such that for all k ∈ N, Up to a further nonrelabeled subsequence we can assume that there exists w ∈ W 1,2 (S n−1 ; R n ) with − S n−1 w = 0 such that w k w in W 1,2 (S n−1 ; R n ), and also pointwise H n−1 -a.e. on S n−1 . But then (recalling (4.22) and (4.23)), since w k → w strongly in L 2 (S n−1 ; R n ) and A(w k ) A(w) weakly in L 2 (S n−1 ; R n ). Thus, by lower semicontinuity of the first two terms in (5.37) under weak convergence in W 1,2 (S n−1 ; R n ), we obtain 0 ≤ Q n, n n−1 (w) ≤ lim inf k→∞ Q n, n n−1 (w k ) = 0 , i.e. Q n, n n−1 (w) = 0 and therefore, by (5.31), i.e., w ∈ H n,1,2 . Moreover, being the (H n−1 -a.e.) pointwise limit of (w k ) k∈N ⊂ H ⊥ n,1,2 , we must also have that w ∈ H ⊥ n,1,2 , since H n,1,2 is finite-dimensional and therefore its orthogonal complement in W 1,2 (S n−1 ; R n ) is a closed subspace. This forces w ≡ 0 on S n−1 and in particular, by (5.42), contradicting the assumption in (5.40) that − S n−1 |∇ T w k | 2 = 1.
Remark 5.11. As we had mentioned in Remark 5.1, the argument just described could have also been used to prove Theorem 1.5 if we knew already that the kernel of Q n (defined in (1.12)) is finite-dimensional (and actually equal to H n,0 ). Nevertheless, as we had remarked therein, there does not seem to be a direct argument to show this fact when n ≥ 4, neither by trying to solve the Euler-Lagrange equation associated to the operator L in (5.1) explicitely, nor by trying to argue as above.
Indeed, if w ∈ ker(Q n ) ⇐⇒ w ∈ ker(Q n,conf )∩ker(Q n,isop ) (see (5.25), (5.26)), then again the following two equations must be satisfied simultaneously, Because of the first equation in (5.44), the second one therein results in the equation For part (i) of the theorem, we have that any map of the form u(x) = Rx with R ∈ SO(n) is of course an orientation-preserving isometry of S n−1 . Conversely, let n ≥ 2, p ∈ [1, ∞] and u ∈ W 1,p (S n−1 ; S n−1 ) be a generalized orientation-preserving isometric map. By definition, this means that at H n−1 -a.e. x ∈ S n−1 the intrinsic gradient of u is an orientation-preserving linear map between T x S n−1 and T u(x) S n−1 , such that or equivalently, in terms of the extrinsic gradient, In particular, for H n−1 -a.e. x ∈ S n−1 one has |∇ T u(x)| 2 n − 1 = 1, and u (ω) = ω for every (n − 1)-form ω on S n−1 . (A.1) By the change of variables formula applied to the vector-valued (n − 1)-form xdv g , we obtain Hence, Poincare's inequality (C.5) on S n−1 , together with (A.1), (A.2), and the fact that |u| ≡ 1, yield As we have also encountered before (see also Appendix C), the equality case in the Poincare inequality implies that in the Fourier expansion of u in spherical harmonics, no other spherical harmonics except the first order ones should appear, hence u(x) = Rx for some R ∈ R n×n . But this linear map would transform S n−1 into the boundary of an ellipsoid, which after possibly an orthogonal change of coordinates is u(S n−1 ) = y = (y 1 , . . . , y n ) ∈ R n : where 0 ≤ α 1 ≤ · · · ≤ α n are the eigenvalues of √ R t R. By assumption, u(S n−1 ) ≡ S n−1 and this forces α 1 = · · · = α n = 1, i.e., R ∈ O(n) and in particular, since u is assumed to be orientation-preserving, For part (ii) we can argue similarly, after making use of the following useful fact.
In the general case of a map u ∈ W 1,n−1 (S n−1 ; S n−1 ) of degree 1, by the approximation property given in [5,Section I.4,Lemma 7], there exists a sequence (u j ) j∈N ⊂ C ∞ (S n−1 ; S n−1 ) with the property that u j → u strongly in W 1,n−1 (S n−1 ; S n−1 ) and degu j = degu = 1 ∀j ∈ N .
Up to passing to a non-relabeled subsequence, we can without loss of generality also suppose that u j → u and ∇ T u j → ∇ T u pointwise H n−1 -a.e. on S n−1 . Since the maps (u j ) j∈N are smooth and surjective, by the previous argument there exist (ξ j ) j∈N ⊂ S n−1 and (λ j ) j∈N ⊂ (0, 1] so that for every j ∈ N, Up to non-relabeled subsequences, we can suppose further that ξ j → ξ 0 ∈ S n−1 and λ j → λ 0 ∈ [0, 1], thus φ ξ j ,λ j → φ ξ 0 ,λ 0 pointwise H n−1 -a.e. on S n−1 , and also weakly in W 1,n−1 (S n−1 ; S n−1 ).

Convergence Theorem and (A.4) is that
Continuing with the proof of Theorem 1.1(ii), we of course have that all the maps given by (1.2) are orientation-preserving conformal diffeomorphisms of S n−1 . Conversely, if u ∈ W 1,n−1 (S n−1 ; S n−1 ) is a generalized orientation-preserving conformal map, similarly to (i) we have that H n−1 -a.e. on S n−1 , or equivalently, in terms of the extrinsic gradient, By taking the determinant in both sides of (A.6), we get that H n−1 -a.e. on S n−1 , Precomposing with the Möbius map φ ξ 0 ,λ 0 ∈ Conf + (S n−1 ) of the previous claim, we have that the map u := u • φ ξ 0 ,λ 0 , whose mean value is 0 by (A.5), is also a generalized orientation-preserving conformal transformation of S n−1 of degree 1, and therefore by (A.7), By approximation, the analytic formula for the degree in terms of integration of (n − 1)-forms on S n−1 holds true forũ as well, i.e., We can now use (A.8), Jensen's inequality, and the sharp Poincare inequality on S n−1 ((C.5)), to obtain the chain of inequalities since − S n−1ũ = 0 and |u| ≡ 1 H n−1 -a.e. on S n−1 . Arguing as in part (i), we deduce thatũ = Rid S n−1 , i.e., u = Rφ ξ,λ , where R ∈ SO(n), ξ := ξ 0 ∈ S n−1 and λ := 1 λ 0 > 0.
Remark A.1. The Möbius transformations of S n−1 given by (1.2) could of course alternatively be described by performing an inversion in T ξ S n−1 with respect to some center, say the origin ξ of the affine hyperplane T ξ S n−1 of R n , and some radius, say √ λ > 0. These maps however would correspond exactly to the Möbius transformations produced by dilation in T ξ S n−1 by factor 1 λ , composed finally with a flip in R n , i.e., an orthogonal map that would change back the orientation. By Liouville's Theorem 1.1, the conformal group of the sphere is given by and is a actually a Lie group, i.e., a differentiable manifold (of dimension n(n+1) 2 ) with a group structure, given by composition of maps. Analytically, the maps (φ ξ,λ ) ξ∈S n−1 ,λ>0 are given by the formula Using (A.10), the corresponding Lie algebra of infinitesimal Möbius transformations, i.e., the tangent space of Conf (S n−1 ) at the id S n−1 , can easily be identified. Indeed, it is then an elementary exercise in differential geometry to check that which is the representation that we made use of in the proof of Lemma 4.13.
Remark A.2. In the conformal case, the argument for the proof of Theorem 1.1(ii) can easily be modified in order to give a compactness statement for sequences of orientation-preserving (resp. orientationreversing) approximately conformal maps on S n−1 of degree 1 (resp. −1), see the subsequent Lemma A.3.
When n = 3, the statement therein essentially reduces to a well known compactness result for harmonic maps of degree ±1 on S 2 (cf. [2, Theorem 2.4, Step 1 in the proof], as well as [18]), which was proven using a concentration-compactness argument in the spirit of P.L. Lions [19]. With the observation that the Möbius transformations can be used to "globally invert" a map from S n−1 to itself, in the sense of fixing its mean value to be 0, we can give a simpler and more elementary proof of this fact, which can be appropriately generalized in every dimension n ≥ 3. For further applications of this simple observation the interested reader is also referred to [15], [25] for two different and shorter proofs of [2,Theorem 2.4]. In the following lemma we present again for simplicity the case of orientation-preserving degree 1 maps.
Lemma A.3. Let n ≥ 3 and (u j ) j∈N ⊂ W 1,n−1 (S n−1 ; S n−1 ) be a sequence of generalized orientationpreserving maps of degree 1 which are approximately conformal, in the sense that which as a condition is in this case equivalent to Then, there exist Möbius transformations (φ j ) j∈N ⊂ Conf + (S n−1 ) and R ∈ SO(n) so that up to a nonrelabeled subsequence, Proof. By the degree 1 condition, as in (A.3),we can again find (ξ j ) j∈N ⊂ S n−1 and (λ j ) j∈N ⊂ (0, 1], so that after setting φ j := φ ξ j ,λ j ∈ Conf + (S n−1 ) andũ j := u j • φ j , we have Because of (A.14) and (A.15), the sequence (ũ j ) j∈N is in particular uniformly bounded in W 1,n−1 (S n−1 ; S n−1 ), hence up to a non-relabeled subsequence converges weakly in W 1,n−1 (S n−1 ; S n−1 ), and up to a further one also pointwise H n−1 -a.e. to a mapũ ∈ W 1,n−1 (S n−1 ; S n−1 ). Sinceũ j →ũ strongly in L n−1 (S n−1 ; S n−1 ), we obtain by (A.14) that in particular, We can then apply the same argument as in (A.9) in the proof of Theorem 1.1(ii), to obtain the chain of and with the same reasoning as in there, we conclude thatũ(x) = Rx for some R ∈ O(n). Finally, by (A.16)-(A.18) we actually obtain thatũ j →ũ strongly in W 1,n−1 (S n−1 ; S n−1 ). Since the degree is stable under this notion of convergence, i.e., indeed R ∈ SO(n).
B Integral identities for Jacobians, Taylor expansions of the deficits and proof of Korn's identity We start this appendix by collecting and proving some integral identities for Jacobians that we used in the bulk of the paper, and especially in the proof of Lemma 3.9.
Lemma B.1. Let u ∈ W 1,2 (S 2 ; R 3 ) and let u h : B 1 → R 3 be as usual its harmonic continuation in B 1 , taken componentwise. Then (with the notation adopted in (1.7) for n = 3), Moreover, if w ∈ W 1,2 (S 2 ; R 3 ) and w h is defined analogously, then where the quadratic form Q V 3 is defined as Remark B.2. As the reader might already know from the theory of null-Lagrangians or notice from the next proof, the above formulas actually hold true with B 1 being replaced by any other open bounded domain U ⊂ R 3 with sufficiently regular boundary, and u h (resp. w h ) being replaced by any other interior extension of u (resp. w), for which the previous bulk integrals are well defined. Since we only used the expressions for the harmonic extension, which is smooth in the interior of the unit ball, we have preferred to state the previous lemma in this particular form.
Proof of Lemma B.1. Regarding the proof of the identity (B.1), the determinant of ∇u h := (∂ j u i h ) 1≤i,j≤3 can be rewritten as where a × b ∈ R 3 denotes the exterior product of two vectors a, b ∈ R 3 . Using the first identity in (B.4), and integrating by parts, we have Here, we have used the vector calculus identities and we have written the full gradients ∇u 2 h and ∇u 3 h on S 2 in terms of the local orthonormal oriented frame {τ 1 , τ 2 , x} for S 2 , which is such that τ 1 × τ 2 = x, τ 2 × x = τ 1 , x × τ 1 = τ 2 . Using the other two expressions from (B.4) and arguing in the same manner, we also obtain and Therefore, summing (B.5)-(B.7), and recalling the notation (1.7) (for n = 3), we arrive at (B.1). Regarding (B.2), the first equality is immmediate from the expansion of the determinant around the identity matrix The expression in the second line of (B.2) follows from the fact that the resulting terms can be written as boundary integrals in the following fashion. Using again Stokes' theorem, we can calculate Subtracting the last two identities we arrive at Hence, the second equality in (B.2) follows from (B.8), (B.9) and (B.1) for w in the place of u here.
Next, we calculate in detail the Taylor expansions up to second order of the geometric quantities that we used in the main body of the paper. The computations presented in the following lemma are formal, and we assume without further clarification that the maps in consideration are always regular enough so that we can perform the expansions. In the case n = 3, we had directly performed the expansion of the 2-Dirichlet energy of u around the id S n−1 in the proof of Lemma 4.5, and the one for V 3 (u) was performed in the previous Lemma B.1. Thus, the focus in the next lemma is mostly on the case n ≥ 4. Moreover, notice that in Subsection 4.2 we had already translated and scaled the initial map u properly (recall (4.2) and (4.3)), which we will also assume for convenience next.
The expansion of the generalized signed-volume V n (u) in (1.6), (1.7) around the id S n−1 , for n = 3 was given in Lemma B.1. An intrinsic way to perform the calculation in every dimension n ≥ 3 is the following.
We have also denoted by (I n,i (w)) i=0,1,2 the zeroth, first and second order terms with respect to w and ∇ T w in the expansion of V n (u) around the id S n−1 respectively, and by I n,3 (w) the remaining term which is a polynomial of order at least 3 and at most n in w and its first derivatives. Keeping in mind that ∂ τ i x = τ i for i = 1, . . . , n − 1 and that by an abuse of notation, τ 1 ∧ · · · ∧ τ n−1 ≡ x, we can compute each term separately.
I n,0 (w) = − S n−1 x, ∂ τ 1 x ∧ · · · ∧ ∂ τ n−1 x = −  x, After integrating by parts it is easy to see that the first term in the last line of (B.26) is − S n−1 1≤i,j≤n−1 ∂ τ i w, τ i ∂ τ j w, τ j = − S n−1 w, (n − 1)(div S n−1 w)x − ∇ T div S n−1 w , (B.27) while the second one therein is − S n−1 1≤i,j≤n−1 ∂ τ i w, τ j ∂ τ j w, τ i = − S n−1 w, (div S n−1 w)x − ∇ T div S n−1 w + (n − 2) where for each k = 2, . . . , n − 1, A n,k is a nonlinear first order differential operator that is a homogeneous polynomial of order k in the first derivatives of w.
Let us conclude this appendix by giving a proof of Korn's identity on S n−1 .
Remark C.1. The following Parseval identities on S n−1 hold true: If u ∈ W 1,2 (S n−1 ; R n ) with its Fourier expansion in spherical harmonics being u = ∞ k=0 G n,k j=1 α n,k,j ψ n,k,j , then The sharp Poincare inequality for maps u ∈ W 1,2 (S n−1 ; R n ) is then easily deduced. Since λ n,k ≥ n − 1 for every k ≥ 1, from (C.4) we obtain Of course, depending on the number of vanishing first Fourier modes in the expansion of u, the constant in the above inequality can be improved in an obvious way. By expanding a function in spherical harmonics one can often obtain useful estimates. In the next lemma, we mention two of them that we have used earlier in the paper.
Lemma C.2. If u ∈ W 1,2 (S n−1 ; R n ), and u h : B 1 → R n denotes its (componentwise) harmonic extension, the following estimates hold true: Proof. Let us give the proof of these two simple estimates in the case that u is scalar-valued, the case of vector-valued u being an immediate consequence. We write again u = ∞ k=0 G n,k j=1 α n,k,j ψ n,k,j , and therefore, its harmonic extension can be written in polar coordinates (r, θ) ∈ [0, 1] × S n−1 as u h (r, θ) = ∞ k=0 G n,k j=1 r k α n,k,j ψ n,k,j (θ) .
Since 1 n−1 ≤ k k+n−2 ≤ 1 for every k ≥ 1, the desired estimates follows immediately by the above identities and (C.4).