In 1850, Liouville [22] proved that, given a domain Ω ⊂ R 3 , any smooth map u : Ω → R 3 satisfying the differential inclusion Du(x) ∈ R + O(n) for all x ∈ Ω must be either affine or a Möbius transform. A corollary to Liouville's Theorem is that a C 3 function whose gradient belongs everywhere to SO(n) is an affine mapping. This phenomenon of being able to globally control a map satisfying a certain differential inclusion Du ∈ K is known as "rigidity". Questions about the stability of differential inclusions under weak convergence and approximate rigidity statements, raised by Tartar in [30,31], are intimately linked with phenomena of compensated compactness and have been extremely influential in the development of weak convergence methods in PDE.
Here we are interested in quantitative versions of approximate rigidity. In [14] Friesecke, James and Müller solved a long standing open problem by proving an optimal quantitative rigidity estimate for K = SO(n). Specifically, they showed that for every bounded Lipschitz domain Ω ⊂ R n , n ≥ 2, there exists a constant C(Ω) such that, for K = SO(n),
Here and below, dist(M, K) denotes the distance from a matrix M ∈ R n×n to a subset K ⊂ R n×n measured in the Euclidean norm. This result strengthened earlier work of a series of authors, including John [17], Rešetnjak [26], and Kohn [19], and it has had a number of important applications, in particular to thin film limits of elastic structures [14,15]. A number of works have extended the above result (1) to cover various larger classes of matrices than K = SO(n). Chaudhuri and Müller [8] and later De Lellis and Székelyhidi [9] considered a set of the form K = SO(n)A∪SO(n)B where A and B are strongly incompatible in the sense of Matos [24]. Faraco and Zhong [13] proved an analogous quantitative rigidity result with K = m•SO(n) where m ⊂ (0, +∞) is compact. There the infimum in the left-hand side of (1) also needs to include the gradients of Möbius transforms, and the integral is over a smaller subset Ω ⊂⊂ Ω. Similar results for maps defined on the sphere have been obtained recently by Luckhaus and Zemas [23]. Best constants for (1) are investigated by Lewicka and Müller in [21].
Our main result is an optimal generalization of the quantitative rigidity estimate of [14] in the context of compact connected submanifolds K ⊂ R 2×2 without boundary.
Theorem 1.1. Let K ⊂ R 2×2 be a smooth, compact and connected 1-manifold without boundary. Assume that K has no rank-one connections, and satisfies the stronger property of being elliptic in the sense that there exists C * > 0 such that
Then for any u ∈ H 1 (B 1 ; R 2 ) we have
for some constant C = C(K) > 0.
Remark 1.2. A covering argument as in [13] shows that the estimate (3) in the balls B 1/2 ⊂ B 1 automatically improves to inf
This result is optimal among compact connected submanifolds K ⊂ R 2×2 without boundary for the following reasons:
• First, it is classical that the no-rank-one-connections assumption is necessary for the rigidity of the exact differential inclusion (see e.g. [25,18]).
• Second, ellipticity is necessary for the validity of the linearized version of (3) because non-ellipticity would imply that the tangent space T M K has a rank-one connection for some M ∈ K, and by Remark 2.3 the linearized version of (3) is implied by (3).
• Third, the two previous conditions (no rank-one connections and ellipticity) imply that the connected submanifold without boundary K ⊂ R 2×2 must be of dimension 1 [33,Corollary 3.5 & 3.6].
Moreover, we provide in Section 5 an example showing that the two-dimensional setting is also optimal: there exists an elliptic 1-submanifold K ⊂ R 3×3 without rank-one connection but which contains a so-called T 4 configuration, a well-known obstruction to compactness of sequences {u k } ⊂ H 1 satisfying dist(Du k , K) → 0 in L 2 [7], and therefore to any type of quantitative rigidity estimate. One of our motivations for studying differential inclusion into general submanifolds K ⊂ R 2×2 is our previous work [20] where we obtained a rigidity result for a non-elliptic differential inclusion related to the so-called Aviles-Giga functional, and pointed out the nice consequences that a corresponding quantitative rigidity estimate would have. Theorem 1.1 is not valid for non-elliptic differential inclusions, but the ideas in the present work should be relevant to attain that goal.
While the statements of the quantitative rigidity results of [14,8,13] are elementary, their proofs are not. Their starting point, in addition to rigidity of the exact differential inclusion, is a linearized version of (1) for the differential inclusion Du ∈ T M 0 K into a tangent space T M 0 K. For K = SO(n) and M 0 = I, this is Korn's inequality. A natural linearization procedure then provides a quantitative rigidity estimate, but in terms of the L ∞ norm of dist(Du, K), rather than L 2 . Strengthening the L 2 bound on dist(Du, K) into an L ∞ bound constitutes therefore the main difficulty. A key idea, introduced in [14], is to use the regularity of an elliptic PDE satisfied by solutions of the exact differential inclusion: for K = SO(n) the exact differential inclusion Du ∈ SO (n) implies that the coordinate functions u k are harmonic. For K ⊂ R + SO (n) the coordinate functions satisfy the (n -2)-Laplace equation div(|∇u k | n-2 ∇u k ) = 0. Such PDE follows from the universal identity div cof(Du) = 0 (where cof denotes the cofactor matrix), together with identities satisfied by matrices in the specific set K. It is satisfied by solutions of the exact differential inclusion, and for a general map u the error from solving that PDE can be controlled in terms of the right-hand side of (1). This allows to reduce the proof of (1) to maps solving that PDE. Elliptic regularity then provides, via a compactness argument, a uniform bound on dist(Du, K) and the linearization can be performed.
Following this scheme, the main ingredient to prove Theorem 1.1 is to embed K into the graph of a uniformly monotone vector field: this will be enough to turn the identity div cof(Du) = 0 into a quasilinear elliptic equation for the exact differential inclusion Du ∈ K. Proposition 1.3. Let K be as in Theorem 1.1. There exist G 1 , G 2 : R 2 → R 2 smooth, globally Lipschitz vector fields such that
and G 1 , G 2 are uniformly monotone, that is
for some constant λ > 0 depending only on K.
Proposition 1.3 relies on remarkable properties of elliptic subsets of R 2×2 with respect to the decomposition into conformal and anticonformal parts, discovered in [33] and exploited in a striking manner in [5,11,12] (see also [18]). It is also related to the classical link between two-dimensional elliptic PDEs of second order and complex Beltrami equations, see e.g. the introduction of [6], and [4,2,1,3] for recent developments on nonlinear Beltrami equations. The proof of Proposition 1.3 is in fact extremely close to [3,Theorem 5], and the main point of Proposition 1.3 is to emphasize that the second order elliptic equations satisfied by the real and imaginary part of solutions to a nonlinear Beltrami equation (as in [3,Theorem 5]) are, in our setting, associated to embeddings of K into graphs: this fact is crucially used in Step 2 of Theorem 1.1's proof, and we find it convenient to state it independently (although it could also be retrieved from the proof of [3,Theorem 5] applied to the nonlinear Beltrami equation associated to K). Then the proof of Theorem 1.1 follows the scheme outlined above.
The article is organized as follows. In Section 2 we establish the two basic prerequisites to Theorem 1, rigidity for the exact differential inclusion and the linearized estimate. In Section 3 we give the proof of Proposition 1.3. In Section 4 we gather these ingredients to prove Theorem 1.1. In Section 5 we describe the counterexample in R 3×3 .
In this section, let K ⊂ R 2×2 be as in Theorem 1.1. We prove the following two Lemmas.
) is such that Du ∈ K a.e., then Du ≡ M for some M ∈ K.
Under the additional assumption that u coincides with an affine map at the boundary, Lemma 2.1 would follow directly from [32,Theorems 2 & 3]. Without restrictions on boundary values, it is a simple consequence of the smoothness result in [32, § 5] and the fact that our set K is a closed one-dimensional curve, as will be clear from the short proof below. Lemma 2.2. For all M ∈ K and u ∈ H 1 (B 1 ; R 2 ) we have
for some constant C = C(K) > 0, where T M K denotes the linear tangent space to K at M .
Remark 2.3. The linearized estimate (4), or rather its weaker interior version
is a necessary condition for (3) to be valid. Assume indeed that (3) is verified, fix u ∈ C 1 (B 1 ; R 2 ), and apply (3
In particular X is bounded, and extracting a converging subsequence we obtain X ∈ T M K showing the validity of ( 5) for u ∈ C 1 (B 1 ; R 2 ), and then by density for u ∈ H 1 (B 1 ; R 2 ).
Proof of Lemma 2.1. Let = |K| and γ : R/ Z → K be an arc-length parametrization of K.
The ellipticity assumption (2)
, then u is smooth by [32], and since B 1 is simply connected there exists a smooth lifting θ :
Using that div cof(Du) = 0, where cof denotes the cofactor matrix, we find cof(γ (θ))∇θ = 0, hence ∇θ = 0 since cof(γ (θ)) is invertible. Therefore Du is constant.
Proof of Lemma 2.2. For M ∈ K we denote by
We denote by P αβjk M ∈ R the coefficients of P M , that is,
so the differential operator u → P M Du has symbol P M (iξ), i.e.,
We claim that P M (ξ) has trivial kernel for all non-zero ξ
are real-valued. In other words, the real and imaginary parts of v ⊗ iξ both belong to ker P M = T M K. Since T M K is a one-dimensional subspace of R 2×2 which doesn't contain any rank-one matrix, we have T M K = RX 0 for some invertible matrix X 0 . Hence we deduce that v ⊗ iξ = λX 0 for some λ ∈ C and an invertible matrix X 0 ∈ T M K. But v ⊗ iξ has zero determinant, so λ = 0 and we must have v = 0. This proves that P M (ξ) has trivial kernel. Therefore we have the representation formula [28, Theorem 4.1] and the coercive inequality [28,Theorem 8.15] that follows from it,
for all u ∈ H 1 (B 1 ; R 2 ). (In the notation of [28], N = 4, M = 2, and the index set {1, 2, 3, 4} for j is in our case given by {1, 2} 2 , and we can take m j = 1 for j ∈ {1, 2} 2 and l i = 0 for i ∈ {1, 2}.) The constant C > 0 in (6) depends a priori on the fixed matrix M ∈ K. Denote by C(M ) the best possible constant in (6). Then for any M, M ∈ K we have
For all M ∈ K, there exists δ(M ) > 0 sufficiently small such that for all
By compactness, we can cover K with a finite collection of balls B δ(M j ) (M j ) : M j ∈ K , hence C(M ) ≤ 4 max{C(M j )} for all M ∈ K, and we can take the constant C in (6) to depend only on K.
Moreover, if u ∈ H 1 (B 1 ; R 2 ) satisfies P M Du = 0 a.e., then Du = λX 0 for some λ ∈ L 2 (B 1 ; R), and the distributional identity 0 = div cof(Du) = cof(X 0 )∇λ implies that λ is constant, hence Du ≡ X for some X ∈ T M K.
Therefore (4) follows from ( 6) via a compactness argument: assume by contradiction the existence of sequences M k ∈ K, and
For any given k, the function X → ´B1 |Du k -X| 2 dx is a strictly convex quadratic polynomial on the finite-dimensional space T M k K, so the infimum in the left-hand side is attained at a unique X k ∈ T M k K. Subtracting from u k its average and X k x, we may in fact assume
Thus we may extract subsequences (not relabeled)
), and thus by lower semicontinuity of the L 2 norm under weak convergence, we have P M Du = 0 a.e., which implies Du ≡ X for some
, and thus Du ≡ X = 0. Further u satisfies ´B1 u dx = 0, which implies u ≡ 0. Plugging u k into (6) gives
Passing to the limit as k → ∞ and using the strong L 2 convergence, we have 1 ≤ C ´B1 |u| 2 dx = 0, which gives a contradiction.
Remark 2.4. The fact that (4) follows from ( 6) is general and already mentioned in [28, p.74], here we provided details for the reader's convenience. More precisely, if P M D is replaced by a more general differential operator, the validity of ( 4) is equivalent to the null space of that differential operator being finite-dimensional [28, Theorem 8.15 & Remark 4], and a compactness argument similar to the one given here shows that the last term in the right-hand side of ( 6) can be dropped if u is orthogonal to that finite-dimensional null space.
3 Proof of Proposition 1.3
We only prove the existence of G 1 , the existence of G 2 is obtained by the same arguments. The proof relies on the properties of the conformal and anticonformal projections of K uncovered in [33,12]. For any z + , z -∈ C, we denote by
the 2 × 2 matrix whose conformal, respectively anticonformal, part is represented by z + , respectively z -. For any A ∈ R 2×2 , the decomposition A = [z + , z -] is unique, and we have the identities
where |A| and A denote the Hilbert-Schmidt and the operator norms of A, respectively. We denote by p + : [z + , z -] → z + the projection onto the conformal part. Using these notations, the ellipticity assumption ( 2) is equivalent to
for all [z + , z -], [z + , z -] ∈ K, and corresponds exactly to the statement that the curve K is C * -elliptic in the sense of [12,Def. 1]. This condition, as observed in [33, Theorem 3.2], see also [12, Lemma 1], implies that
Proof of Lemma 3.1. We first fix, thanks to Kirszbraun's theorem, a k-Lipschitz extension H : C → C. In the rest of the proof we modify H to make it smooth while still agreeing with H on p + (K), at the cost of slightly increasing its Lipschitz constant.
Therefore we may reparametrize and consider
with z(s) an arc-length parametrization of p + (K) and + its length, and the map s → H(z(s)) is smooth by smoothness of K.
For small enough δ > 0, the map
is a smooth diffeomorphism. We first modify H by setting
where λ is the odd (1 -δ) -1 -Lipschitz function given for r > 0 by
for r > δ.
In particular we have
so H is smooth in U δ 2 (by smoothness of s → H(z(s)) and ϕ -1 ) and agrees with H on p + (K). Note that by definition of ϕ and λ we have Φ(Z) = Z in C \ U δ and therefore Φ is Lipschitz in C. Since Dϕ(s, 0) ∈ SO(2) for all s ∈ R/ + Z, we have Dϕ ≤ 1 + Cδ on R/ + Z × (-2δ, 2δ). Further, we have D(ϕ -1 ) = 1 on p + (K), and D(ϕ -1 ) ≤ 1 + Cδ on U 2δ . Denoting by ψ the (1 -δ) -1 -Lipschitz map (s, r) → (s, λ(r)), we write Φ(Z) = ϕ(ψ(ϕ -1 (Z))) and deduce that DΦ ≤ 1 + Cδ a.e. in U 2δ . This inequality is also true in the rest of C by definition of Φ, so we conclude that H is k-Lipschitz in C, with k = (1 + Cδ)k < 1 for small enough δ > 0. Now δ is fixed and we define, for ∈ (0, δ 2 /4),
for a smooth kernel ρ ≥ 0 with support in B 1 and ´ρ(y) dy = 1, and some smooth cut-off function χ with
, and thus agrees with H on p + (K). Finally, denoting by L the Lipschitz constant of χ, we have
Proof of Lemma 3.2. For any w ∈ C the equation
admits a unique solution z ∈ C thanks to the fixed point theorem, since z → w -H(z) is k-Lipschitz and 0 ≤ k < 1. This shows that F is bijective. The inequalities
follow directly from the fact that H is k-Lipschitz and imply the announced Lipschitz constants of F and F -1 . The inverse F -1 is smooth thanks to the Inverse Function Theorem, since
Proof of Proposition 1.3 completed. Then, identifying C with R 2 , we define
The map G 1 is smooth and globally Lipschitz with Lipschitz constant Λ = (1 + k)/(1 -k).
Moreover, for all A = F (z), A = F (z ), we have
This concludes the proof of Proposition 1.3.
Step 1. We may assume that u is Lipschitz, thanks to the truncation result [14, Proposition A.1]. Part of the statement of [14, Proposition A.1] is that for any Lipschitz domain Ω ⊂ R n and any function ũ ∈ W 1,p (Ω; R m ) (for p ≥ 1), there exists some constant C(Ω, m, p) > 0 such that for any λ > 0 there exists ṽ : Ω → R m satisfying Dṽ L ∞ (Ω) ≤ C(Ω, m, p)λ and Dũ -Dṽ
Then for all X ∈ R 2×2 with |X| > 2R, we have |X| ≤ 2 dist(X, K). Applying [14, Proposition A.1] with λ = 2R gives v :
for some constant C 0 (depending only on B 1 ). If there exists
then repeated applications of the triangle inequality give
Thus if Theorem 1.1 holds for all Lipschitz mappings v for some constant C(K), then it is also valid for all H 1 mappings with
Step 2. We may assume in addition that u
Consider indeed w ∈ C 2 (B 1 ) such that w = u on ∂B 1 and
The existence of such w is guaranteed by the ellipticity of the equation 0 = div G j (Dw j ) = tr(DG j (Dw j )D 2 w j ) = 0, invoking e.g. [16,Theorem 12.5]: the inequality λ|ξ| 2 ≤ DG j (A)ξ • ξ ≤ Λ|ξ| 2 , valid for all A, ξ ∈ R 2 thanks to Proposition 1.3, ensures that the eigenvalues of the symmetric part [DG j (A)] s = (DG j (A) + DG j (A) T )/2 of DG j (A) are bounded above and below (since
) and in particular condition (ii) in [16,Theorem 12.5] is satisfied. Letting v = u -w and using the uniform monotonicity of G 1 we find
Since div G 1 (Dw 1 ) = 0 and div(iDu 2 ) = 0 we rewrite this as
and infer
According to Proposition 1.3 the function M → G 1 (A) + iB, where A, B denote the first and second row of the matrix M , vanishes on K. Since that function is Lipschitz we deduce that |G 1 (Du 1 ) + iDu 2 | ≤ C dist(Du, K), and therefore
Applying a similar argument to v 2 we obtain
Recalling that v = u -w and using the triangle inequality we deduce
As a consequence, if Theorem 1.1 is valid for w then we obtain it for u. This proves Step 2.
Step 3. As u j ∈ C 2 (B 1 ) satisfies ( 8), it is a weak solution of
Invoking e.g. Theorem 1 in [10, § 6.3], we have that
for any i, j ∈ {1, 2}. This combined with the Sobolev embedding theorem implies that
for any α > 0 and some constant C = C(K). Thanks to this estimate and the exact rigidity obtained in Lemma 2.1, we may argue exactly as in [13,Lemma 4.5] to deduce that inf
for some function ρ depending only on K and satisfying ρ( ) → 0 as → 0.
Step 4. We finally combine Step 3 with the linearized estimate of Lemma 2.2, to obtain our main estimate (3). The basic idea, as in [14,13], is to linearize dist 2 (•, K) around M 0 ∈ K such that |Du -M 0 | is uniformly small. When doing so, (3) formally turns into the linearized estimate (4) of Lemma 2.2, and it remains to control the error terms. Due to the modification of u arising from the translation X ∈ T M K in the left-hand side of the linearized estimate (4), it is not directly obvious that the error terms are negligible. In [14] this problem is absent because their equivalent of (9) comes with an explicit ρ( ) = C 1 4 . In [13] it is taken care of via a topological degree argument [13, Proposition 4.7] (see also [27]) which allows to avoid the translation. While a similar degree argument could be used to iteratively improve the estimate here, we present a simpler alternative method relying on elementary estimates.
We assume without loss of generality that
where 0 = 0 (K) is to be chosen in the course of the proof. If (10) is not valid then (3) is automatically satisfied for a large enough constant C because the left-hand side of( 3) is bounded thanks to Step 1. We fix δ 0 > 0 depending only on K, such that the nearest-point projection Π K onto K is uniquely defined and smooth in the neighborhood N 2δ 0 (K). We first choose 0 small enough that ρ( 0 ) ≤ δ 0 , so thanks to (9) the projection Π K (Du) is well-defined.
We claim that, for every M ∈ K, there exists
Here and in the rest of this proof we denote by C > 0 a generic constant depending only on K.
To prove (11), we first invoke Lemma 2.2, according to which we have
Choosing X = X M ∈ T M K attaining the infimum in the left-hand side, we obtain
Moreover the minimizing property of X M implies that the function t → ´B1/2 |Du -M -tX M | 2 dx has zero derivative at t = 1 and hence ´B1/2 (Du -M -X M ) dx is orthogonal to X M . We deduce
and therefore
Recalling from the proof of Lemma 2.2 that P M denotes the orthogonal projection onto (T M K) ⊥ , we estimate the integrand in the right-hand side of (12) using
The last inequality follows from the fact that
Now we may choose Y M ∈ K such that
Indeed, if |X M | ≤ δ 0 then one can simply take Y M = Π K (M + X M ) and use the fact that DΠ K (M )X M = (I -P M )X M = X M , and if |X M | ≥ δ 0 one may take Y M = M . From ( 14) and ( 15) we infer
Using (13) and Cauchy-Schwarz to estimate the last term, we deduce (11).
Next we want to discard the last term in the right-hand side of (11). To that end we first fix, thanks to ( 9)-( 10), an
and Y = M 0 provides therefore a better choice to optimize the infimum in the left-hand side of (16). Moreover, applying (11) to any
, and we updated the value of the generic constant C. Since this is valid for all
Recalling (16), this implies
Since ρ( ) → 0 as → 0, we may choose 0 such that Cρ( 0 ) 2 ≤ 1/2, and absorb the last term in the left-hand side, thus proving (3).
In this section we prove that the two-dimensional setting of Theorem 1.1 is optimal in the following sense: a connected 1-submanifold of R 3×3 which has no rank-one connection and is elliptic may not satisfy Šverák's compactness result [32], and even less a quantitative rigidity estimate.
We recall (see e.g. [29, § 2]) that an ordered set of N ≥ 4 matrices {T i } N i=1 ⊂ R m×n without rank-one connections is said to form a T N configuration if there exist matrices P i , C i ∈ R m×n and numbers κ i > 1 such that
. . .
where C i is rank-one for all i and N i=1 C i = 0.
Proposition 5.1. There exists a smooth, compact and connected 1-submanifold K ⊂ R 3×3 without boundary which is elliptic and has no rank-one connection, but contains a T 4 configuration.
By a known construction, see e.g. [7, Theorem 3.1], Proposition 5.1 implies the existence of a sequence of maps u k : R 3 → R 3 such that ˆB1 dist 2 (Du k , K) dx → 0 as k → 0, but (Du k ) is not precompact in L 2 (B 1/2 ). In particular, one certainly cannot hope for a quantitative estimate
for any function ρ( ) → 0 as → 0.
Proposition 5.1 is a consequence of the construction below. Let a > 0 and define matrices T 1 , T 2 , T 3 , T 4 by
We have that T k -C k is rank-one for k = 1, 2, 3, 4, and (with the convention that C 5 = C 1 ) Let ρ : R → R be a smooth monotonically increasing function to be determined later that satisfies
Then we have
so K a contains the T 4 configuration {T 1 , T 2 , T 3 , T 4 }. Next we adjust the parameter a > 0 and the function ρ in order to ensure that K a has no rank-one connection and is elliptic.
Notation. With the M i 1 i 2 ,j 1 j 2 minor we mean the determinant of the 2 × 2 submatrix corresponding to the rows i 1 , i 2 and columns j 1 , j 2 .
Lemma 5.2. If a > 0 is such that θ a / ∈ π 48 Z, the curve K a is elliptic, i.e. Rank Γ a (θ) > 1 for all θ ∈ R. We assume that there exist θ
and we obtain a contradiction. We do this in several steps.
Step 1. We have
Proof of Step 1. From ( 17) calculating the M 12,12 minor we have
And calculating the M 23,12 minor we have
Adding and subtracting (20) and ( 21) we obtain the equations 0 = sin 3 θ -θ sin 4 θ -θ cos 3 θ + θ -6θ a cos 4 θ + θ -8θ a (22) and
Since θ = θ in R/2πZ, the last factor of ( 23) is nonzero, so either the first or the second must be zero. This implies θ + θ ∈ πZ. As a consequence, the last two factors in (22) are equal to ± cos(6θ a ) and cos(8θ a ) and are nonzero by our choice of a. So one of the first two factors of ( 22) must vanish, that is, θ -θ ∈ π 3 Z ∪ π 4 Z. From Step 1 we have θ + θ ∈ πZ so the second factor is cos(8θ a ) = 0. The last factor is arbitrarily close to ± cos(6θ a ) = 0 since |ρ-id| ≤ . The third factor is nonzero by construction of ρ (recall Lemma 5.3) because θ, θ ∈ π 24 Z by (19). So we must have sin(4(θ -θ )) = 0 hence θ -θ ∈ π 4 Z.
Since θ -θ ∈ π 4 Z and |ρ -id| ≤ , the third factor in the left-hand side of (24) has absolute value ≤ 4 . Since θ + θ ∈ πZ, the second factor in the right-hand side of ( 24) has absolute value equal to | cos(6θ a )| > 0, and for small enough the absolute value of the last factor is ≥ | cos(6θ a )|/2 > 0. Taking also into account that the first and third factors in the right-hand side of (24) differ from each other by an error ≤ 3 , we must have sin 2 3 θ -θ ≤ c a , for some c a > 0 depending only on a. Because θ -θ ∈ π 4 Z by Step 2, provided is chosen small enough, this implies θ -θ ∈ π 3 Z ∩ π 4 Z = πZ.
Step 4: Conclusion. From Step 1 and Step 3 we have θ + θ , θ -θ ∈ πZ, so θ, θ ∈ π 2 Z. Since me may without loss of generality exchange the roles of θ and θ and choose arbitrary representants in R/2πZ, this amounts to θ = 0 and θ = π, or θ = π/2 and θ = 3π/2. In the former case, considering the M 12,13 minor of Γ a (0) -Γ a (π) as in (24) we deduce sin(4(ρ(π) -ρ(0))) cos(4(ρ(π) + ρ(0)) -8θ a ) = 0.
Using again that |ρ -id| ≤ , the second factor has absolute value ≥ | cos(8θ a )|/2 provided is small enough, and the first factor is nonzero by construction of ρ, so we conclude that (18) is not possible for θ = θ ∈ R/2πZ. In the latter case, considering the M 12,23 minor of Γ a (π/2) -Γ a (3π/2) and following similar calculations as in (24) leads to sin(3(ρ(π/2) -ρ(3π/2))) cos(3(ρ(π/2) + ρ(3π/2)) -6θ a ) = 0.
Finally, we follow exactly the same lines as above to get a contradiction.