Nonminimality of spirals in sub-Riemannian manifolds

We show that in analytic sub-Riemannian manifolds of rank 2 satisfying a commutativity condition spiral-like curves are not length minimizing near the center of the spiral. The proof relies upon the delicate construction of a competing curve.


Introduction
The regularity of geodesics (length-minimizing curves) in sub-Riemannian geometry is an open problem since forty years. Its difficulty is due to the presence of singular (or abnormal) extremals, i.e., curves where the differential of the end-point map is singular (it is not surjective). There exist singular curves that are as a matter of fact length-minimizing. The first example was discovered in [9] and other classes of examples (regular abnormal extremals) are studied in [13]. All such examples are smooth curves.
When the end-point map is singular, it is not possible to deduce the Euler-Lagrange equations with their regularizing effect for minimizers constrained on a nonsmooth set. On the other hand, in the case of singular extremals the necessary conditions given by Optimal Control Theory (Pontryagin Maximum Principle) do not provide in general any further regularity beyond the starting one, absolute continuity or Lipschitz continuity of the curve.
The most elementary kind of singularity for a Lipschitz curve is of the corner-type: at a given point, the curve has a left and a right tangent that are linearly independent. In [8] and [3] it was proved that length minimizers cannot have singular points of this kind. These results have been improved in [11]: at any point, the tangent cone to a length-minimizing curve contains at least one line (a half line, for extreme points), see also [4]. The uniqueness of this tangent line for length minimizers is an open problem. Indeed, there exist other types of singularities related to the non-uniqueness of the tangent. In particular, there exist spiral-like curves whose tangent cone at the center contains many and in fact all tangent lines, see Example 2.5 below. These curves may appear as Goh extremals in Carnot groups, see [6] and [7,Section 5]. For these reasons, the results of [11] are not enough to prove the nonminimality of spiral-like extremals. Goal of this paper is to show that curves with this kind of singularity are not length-minimizing.
Our notion of horizontal spiral in a sub-Riemannian manifold of rank 2 is fixed in Definition 2.4. We will show that spirals are not length-minimizing when the horizontal distribution D satisfies the following commutativity condition. Fix two vector fields X 1 , X 2 ∈ D that are linearly independent at some point p ∈ M. For k ∈ N and for a multi-index J = (j 1 , . . . , j k ), with j i ∈ {1, 2}, we denote by X J = [X j 1 , [. . . , [X j k−1 , X j k ] · · · ]] the iterated commutator associated with J. We define its length as len(X J ) = k. Let D k (p) be the R-linear span of {X J (p) | len(X J ) ≤ k} ⊂ T p M. In a neighborhood of the center of the spiral, we will assume the following condition Our main result is the following Differently from [8,3,11,4] and similarly to [10], the proof of this theorem cannot be reduced to the case of Carnot groups, the infinitesimal models of equiregular sub-Riemanian manifolds. This is because the blow-up of the spiral could be a horizontal line, that is indeed length-minimizing.
The nonminimality of spirals combined with the necessary conditions given by Pontryagin Maximum Principle is likely to give new regularity results on classes of sub-Riemannian manifolds, in the spirit of [1]. We think, however, that the main interest of Theorem 1.1 is in the deeper understanding that it provides on the loss of minimality caused by singularities.
The proof of Theorem 1.1 consists in constructing a competing curve shorter than the spiral. The construction uses exponential coordinates of the second type and our first step is a review of Hermes' theorem on the structure of vector-fields in such coordinates. In this situation, the commutativity condition (1.2) has a clear meaning explained in Theorem 2.2, that may be of independent interest. Even though our definition of "horizontal spiral" is given in coordinates of the second type, see Definition 2.4, it is actually coordinates-independent, see Remark 2.6.
In Section 3, we start the construction of the competing curve. Here we use the specific structure of a spiral. The gain of length is obtained by cutting one spire near the center. The adjustment of the end-point will be obtained modifying the spiral in a certain number of locations adding "devices" depending on a set of parameters. The horizontal coordinates of the spiral are a planar curve intersecting the positive x 1 -axis infinitely many times. The possibility of adding devices at such locations arbitrarily close to the origin will be a crucial fact.
In Section 4, we develop an integral calculus on monomials that is used to estimate the effect of cut and devices on the end-point of the modified spiral. Then, in Section 5, we fix the parameters of the devices in such a way that the end-point of the modified curve coincides with the end-point of the spiral. This is done in Theorem 5.1 by a linearization argument. Sections 3-5 contain the technical core of the paper.
We use the specific structure of the length-functional in Section 6, where we prove that the modified curve is shorter than the spiral, provided that the cut is sufficiently close to the origin. This will be the conclusion of the proof of Theorem 1.1.
We briefly comment on the assumptions made in Theorem 1.1. The analyticity of M and D is needed only in Section 2. In the analytic case, it is known that length-minimizers are smooth in an open and dense set, see [12]. See also [2] for a C 1 -regularity result when M is an analytic manifold of dimension 3.
The assumption that the distribution D has rank 2 is natural when considering horizontal spirals. When the rank is higher there is room for more complicated singularities in the horizontal coordinates, raising challenging questions about the regularity problem.
Dropping the commutativity assumption (1.2) is a major technical problem: getting sharp estimates from below for the effect produced by cut and devices on the endpoint seems extremely difficult when the coefficients of the horizontal vector fields depend also on nonhorizontal coordinates, see Remark 4.3.

Exponential coordinates at the center of the spiral
In this section, we introduce in M exponential coordinates of the second type centered at a point p ∈ M, that will be the center of the spiral.
Let X 1 , X 2 ∈ D be linearly independent at p. Since the distribution D is bracketgenerating we can find vector-fields X 3 , . . . , X n , with n = dim(M), such that each X i is an iterated commutator of X 1 , X 2 with length w i = len(X i ), i = 3, . . . , n, and such that X 1 , . . . , X n at p are a basis for T p M. By continuity, there exists an open neighborhood U of p such that X 1 (q), . . . , X n (q) form a basis for T q M, for any q ∈ U. We call X 1 , . . . , X n a stratified basis of vector-fields in M.
Let ϕ ∈ C ∞ (U; R n ) be a chart such that ϕ(p) = 0 and ϕ(U) = V , with V ⊂ R n open neighborhood of 0 ∈ R n . Then X 1 = ϕ * X 1 , . . . , X n = ϕ * X n is a system of point-wise linearly independent vector fields in V ⊂ R n . Since our problem has a local nature, we can without loss of generality assume that M = V = R n and p = 0.
After these identifications, we have a stratified basis of vector-fields X 1 , . . . , X n in R n . We say that x = (x 1 , . . . , x n ) ∈ R n are exponential coordinates of the second type associated with the vector fields X 1 , . . . , X n if we have We are using the notation Φ X s = exp(sX), s ∈ R, to denote the flow of a vector-field X. From now on, we assume that X 1 , . . . , X n are complete and induce exponential coordinates of the second type.
We define the homogeneous degree of the coordinate x i of R n as w i = len(X i ). We introduce the 1-parameter group of dilations δ λ : R n → R n , λ > 0, x ∈ R n , and we say that a function f : for all x ∈ R n and λ > 0. An example of δ-homogeneous function of degree 1 is the pseudo-norm The following theorem is proved in [5] in the case of general rank.
Theorem 2.1. Let D = span{X 1 , X 2 } ⊂ T M be an analytic distribution of rank 2.
In exponential coordinates of the second type around a point p ∈ M identified with 0 ∈ R n , the vector fields X 1 and X 2 have the form The analytic functions a j ∈ C ∞ (U), j = 3, . . . , n, have the structure a j = p j + r j , where: (i) p j are δ-homogeneous polynomials of degree w j −1 such that p j (0, x 2 , . . . , x n ) = 0; (ii) r j ∈ C ∞ (U) are analytic functions such that, for some constants C 1 , C 2 > 0 and for x ∈ U, Proof. The proof that a j = p j + r j where p j are polynomials as in (i) and the remainders r j are real-analytic functions such that r j (0) = 0 can be found in [5]. The proof of (ii) is also implicitly contained in [5]. Here, we add some details. The Taylor series of r j has the form where A ℓ = {α ∈ N n : α 1 w 1 + . . . + α n w n = ℓ}, x α = x α 1 1 · · · x αn n and c αℓ ∈ R are constants. Here and in the following, N = {0, 1, 2, . . .}. The series converges absolutely in a small homogeneous cube Q δ = {x ∈ R n : x ≤ δ} for some δ > 0, and in particular The estimate for the derivatives of r j is analogous. Indeed, we have Thus the leading term in the series has homogeneous degree w j − w i and repeating the argument above we get the estimate When the distribution D satisfies the commutativity assumption (1.2) the coefficients a j appearing in the vector-field X 2 in (2.3) enjoy additional properties.
, where x ∈ R n and t ∈ R. Here, we are using the exponential coordinates (2.1). In the following we omit the composition sign •. Defining Θ : . We claim that there exists a C > 0 independent of t such that, for t → 0, We will prove claim (2.5) in Lemma 2.3 below. From (2.5) it follows that there exist mappings R t ∈ C ∞ (R n , R n ) such that and such that |R t | ≤ Ct 2 for t → 0. By the structure (2.3) of the vector fields X 1 and X 2 and since Θ t, By (1.2), from (2.6) and (2.7) we obtain and we conclude that Thus the coefficients a j (x 1 , . . , n, depend only on the first two variables, completing the proof. In the following lemma, we prove our claim (2.5). Proof. Let X = X j for any j = 3, . . . , n and define the map is the identity and thus T X 0,x 1 ,x 2 ;s = 0. So, claim (2.5) follows as soon as we show thaṫ for any s ∈ R and for all x 1 , x 2 ∈ R. We first compute the derivative of Θ t,x 1 ,x 2 with respect to t. Letting Ψ t, ] with X 1 appearing ν times and c ν, In order to prove thatṪ X 0,x 1 ,x 2 ;s vanishes for all x 1 , x 2 and s, we have to show that for any ν ≥ 1 and for any x 2 and s. From Φ X 0 = id it follows that g(x 2 , 0) = 0. Then, our claim (2.9) is implied by Actually, this is a Lie derivative and, namely, In a similar way, for any k ∈ N we have with X 2 appearing k times. Since the function x 2 → h(x 2 , s) is analytic our claim (2.10) follows.
Without loss of generality, we shall focus our attention on spirals that are oriented clock-wise, i.e., with a phase satisfying ϕ(t) → ∞ andφ(t) → −∞ as t → 0 + . Such a phase is decreasing near 0. Notice that if ϕ(t) → ∞ andφ(t) has a limit as t → 0 + then this limit must be −∞.
Example 2.5. An interesting example of horizontal spiral is the double-logarithm spiral, the horizontal lift of the curve κ in the plane of the form (2.12) with phase ϕ(t) = log(− log t), t ∈ (0, 1/2]. In this case, we havė and clearly ϕ(t) → ∞ andφ(t) → −∞ as t → 0 + . In fact, we also have tφ ∈ L ∞ (0, 1/2), which means that κ and thus γ is Lipschitz continuous. This spiral has the following additional properties: i) for any v ∈ R 2 with |v| = 1 there exists an infinitesimal sequence of positive real numbers (λ n ) n∈N such that κ(λ n t)/λ n → tv locally uniformly, as n → ∞; ii) for any infinitesimal sequence of positive real numbers (λ n ) n∈N there exists a subsequence and a v ∈ R 2 with |v| = 1 such that κ(λ n k t)/λ n k → tv as k → ∞, locally uniformly.
This means that the tangent cone of κ at t = 0 consists of all half-lines in R 2 emanating from 0.
Remark 2.6. We show that Definition 2.4 of a horizontal spiral does not in fact depend on the chosen coordinates.

Cut and correction devices
In this section, we begin the construction of the competing curve. Let γ be a spiral with horizontal coordinates κ as in (2.12). We can assume that ϕ is decreasing and that ϕ(1) = 1 and we denote by ψ : [1, ∞) → (0, 1] the inverse function of ϕ. For k ∈ N and η ∈ [0, 2π) we define t kη ∈ (0, 1] as the unique solution to the equation ϕ(t kη ) = 2πk + η, i.e., we let t kη = ψ(2πk + η). The times will play a special role in our construction. The points κ(t k ) are in the positive x 1 -axis. For a fixed k ∈ N, we cut the curve κ in the interval [t k+1 , t k ] following the line segment joining κ(t k+1 ) to κ(t k ) instead of the path κ, while we leave unchanged the remaining part of the path. We call this new curve κ cut k and, namely, we let We denote by γ cut k ∈ AC([0, 1]; M) the horizontal curve with horizontal coordinates κ cut k and such that γ cut To correct the errors produced by the cut on the end-point, we modify the curve κ cut k using a certain number of devices. The construction is made by induction.
In the lifting formula (2.11), the intervals whereγ 2 = 0 do not contribute to the integral. For this reason, in (3.2) we may cancel the second and fourth lines, wherė D 2 (γ; E ) = 0, and then reparameterize the curve on [0, 1]. Namely, we define the discontinuous curve D(κ; E ) : [0, 1] → R 2 as and then we consider the "formal" i-th coordinate The following identities can be checked by an elementary computation (for ε > 0) (3.4) With this notation, the final error produced on the i-th coordinate by the correction device E is (3.5) The proof of this formula is elementary and can be omitted. We will iterate the above construction a certain number of times depending on a collections of triples E . We first fix the number of triples and iterations.
For i = 3, . . . , n, let B i = {(α, β) ∈ N 2 : α + β = w i − 2}, where w i ≥ 2 is the homogeneous degree of the coordinate x i . Then, the polynomials p i given by Theorem 2.1 and Theorem 2.2 are of the form for suitable constants c αβ ∈ R. We set and we consider an (ℓ − 2)-tuple of triplesĒ = (E 3 , . . . , E ℓ ) such that h ℓ < h ℓ−1 < . . . < h 3 < k. Each triple is used to correct one monomial. Without loss of generality, we simplify the construction in the following way. In the sum (3.6), we can assume that c αβ = 0 for all (α, β) ∈ B i but one. Namely, we can assume that and with c α i β i = 1. In this case, we have ℓ = n and we will use n − 2 devices associated with the triples E 3 , . . . , E n to correct the coordinates i = 3, . . . , n. By the bracket generating property of the vector fields X 1 and X 2 and by the stratified basis property for X 1 , . . . , X n , the pairs (α i , β i ) satisfy the following condition From now on in the rest of the paper we will assume that the polynomials p i are of the form (3.8) with (3.9). Now we clarify the inductive step of our construction. Let E 3 = (h 3 , η 3 , ε 3 ) be a triple such that h 3 < k. We define the curve κ (3) = D(κ cut k ; E 3 ). Given a triple E 4 = (h 4 , η 4 , ε 4 ) with h 4 < h 3 we then define κ (4) = D(κ (3) ; E 4 ). By induction on ℓ ∈ N, given a triple E ℓ = (h ℓ , η ℓ , ε ℓ ) with h ℓ < h ℓ−1 , we define κ (ℓ) = D(κ (ℓ−1) ; E ℓ ). When ℓ = n we stop.
We define the planar curve D(κ; k,Ē ) ∈ AC([0, 1 + 2ε]; R 2 ) as D(κ; k,Ē ) = κ (n) according to the inductive construction explained above, whereε = |ε 3 | + . . . + |ε n |. Then we call D(γ; k,Ē ) ∈ AC([0, 1 + 2ε]; M), the horizontal lift of D(κ; k,Ē ) with D(γ; k, E )(0) = γ(0), the modified curve of γ associated withĒ and with cut of parameter k ∈ N. There is a last adjustment to do. In [0, 1 + 2ε] there are 2(n − 2) subintervals whereκ  Our next task is to compute the error produced by cut and devices on the end-point of the spiral. For i = 3, . . . , n and for t ∈ [0, 1] we let When t < t k+1 or t > t k we haveκ 2 =κ 2 and so the definition above reads ∆ γ i (t) = a i (κ(t)) − a i (κ(t)) κ 2 (t). By the recursive application of the argument used to obtain (3.5), we get the following formula for the error at the final timet = t hn : (3.11) In (3.11) and in the following, we use the following notation for the intervals: with t h 2 = t k . We used also the fact that on [0, t k+1 ] we have γ =γ. On the interval F k we haveκ 2 = 0 and thus On the intervals A j we have κ =κ and thus A j ∆ γ i dt = 0, (3.14) because the functions a i depend only on κ. Finally, on the intervals B j we havē κ 1 = κ 1 + ε j and κ 2 =κ 2 and thus Our goal is to find k ∈ N and devicesĒ such that E k,Ē i = 0 for all i = 3, . . . , n and such that the modified curve D(γ; k,Ē ) is shorter than γ.

Effect of cut and devices on monomials and remainders
Let γ be a horizontal spiral with horizontal coordinates κ ∈ AC([0, 1]; R 2 ) of the form (2.12). We prove some estimates about the integrals of the polynomials (3.8) along the curve κ. These estimates are preliminary to the study of the errors introduced in (3.11).
For α, β ∈ N, we associate with the monomial p αβ (x 1 , When p i = p αβ , the function γ αβ is the leading term in the i-th coordinate of γ in exponential coordinates. In this case, the problem of estimating γ i (t) reduces to the estimate of integrals of the form where ω ≤ η are angles, t ω = ψ(ω) and t η = ψ(η). These integrals are related to the integrals In the following, we will use the short notation D αβ ω = cos α+1 (ω) sin β+1 (ω).
Remark 4.3. We will use the estimates (4.5) in the proof of the solvability of the end-point equations. In particular, the computations above are possible thanks to the structure of the monomials p i : here, their dependence only on the variables x 1 and x 2 , ensured by (1.2), is crucial. When the coefficients a i depend on all the variables x 1 , . . . , x n , repeating the same computations seems difficult. Indeed, in the integrals (4.1) and (4.2) there are also the coordinates γ 3 , . . . , γ n . Then, the new identity (4.3) becomes more complicated because other addends appear after the integration by parts, owing to the derivatives of γ 3 , . . . , γ n . Now, by the presence of these new terms the estimates from below in (4.5) are difficult, while the estimates from above remain possible.
In the study of the polynomial part of integrals in (3.15) we need estimates for the quantities Lemma 4.4. We have where O(ε 2 )/ε 2 is bounded as ε → 0.
Proof. The proof is an elementary computation: (4.7) .
We estimate the terms in (3.13). The quantities ∆ γ i are introduced in (4.6).
Lemma 4.5. Let γ be a horizontal spiral with phase ϕ. For all i = 3, . . . , n and for all k ∈ N large enough we have Proof. By (4.3) with vanishing boundary contributions, we obtain so we are left with the estimate of the integral of r i . Using κ 2 = t sin(ϕ(t)) we get where we let From (2.12), we have |κ(t)| ≤ t for all t ∈ [0, 1]. By part (ii) of Theorem 2.1 we have |r i (x)| ≤ C x w i for all x ∈ R n near 0, with w i = α i + β i + 2. It follows that |r i (κ(t))| ≤ Ct w i for all t ∈ [0, 1], and |R i (t)| ≤ Ct w i +1 . We deduce that and the claim follows. Now we study the integrals in (3.15). Let us introduce the following notation Lemma 4.6. Let γ be a horizontal spiral with phase ϕ. Then for any j = 3, . . . , n and for |ε j | < t h j η j , we have where C > 0 is constant.
Proof. For t ∈ B j we have κ 2 (t) =κ 2 (t) andκ 1 (t) = κ 1 (t)+ε j . By Lagrange Theorem it follows that δ γ r i (t) = ε j ∂ 1 r i (κ * (t)), where κ * (t) = (κ * 1 (t), κ 2 (t)) and κ * 1 (t) = κ 1 (t) + δ j , 0 < δ j < ε j . By Theorem 2.1 we have |∂ 1 r i (x)| ≤ C x w i −1 and so, also using δ j < ε j < t, This implies |δ γ r i (t)| ≤ C|ε j |t w i −1 . Now, the integral we have to study is We integrate by parts the integral withoutφ, getting Since the boundary term is 0, we obtain Remark 4.7. We stress again the fact that, when the coefficients a i depend on all the variables x 1 , . . . , x n , the computations above become less clear. As a matter of fact, there is a non-commutative effect of the devices due to the varying coordinates γ 3 , . . . , γ n that modifies the coefficients of the parameters ε j .

Solution to the end-point equations
In this section we solve the system of equations E k,Ē i = 0, i = 3, . . . , n. The homogeneous polynomials p j are of the form p j (x 1 , The quantities (3.13), (3.14) and (3.15) are, respectively, where we used the short-notation I α i β i k = I α i β i 2πk,2π(k+1) . So the end-point equations We will regard k, h j , and η j as parameters and we will solve the system of equations (5.2) in the unknowns ε = (ε 3 , . . . , ε n ). The functions f i : R n−2 → R are analytic and the data b i are estimated from above by (4.8): Theorem 5.1. There exist real parameters η 3 , . . . , η n > 0 and integers h 3 > . . . > h n such that for all k ∈ N large enough the system of equations (5.2) has a unique solution ε = (ε 3 , . . . , ε n ) satisfying for a constant C > 0 independent of k.
Proof. We will use the inverse function theorem. Let A = a ij i,j=3,...,n ∈ M n−2 (R) be the Jacobian matrix of f = (f 3 , . . . , f n ) in the variables ε = (ε 3 , . . . , ε n ) computed at ε = 0. By (4.6) and Lemma 4.6 we have Here, we are using the fact that for h j → ∞ we have The proof of Theorem 5.1 will be complete if we show that the matrix A is invertible. We claim that there exist real parameters η 3 , . . . , η n > 0 and positive integers h 3 > . . . > h n such that det(A) = 0.

Nonminimality of the spiral
In this section we prove Theorem 1.1. Let γ ∈ AC([0, 1]; M) be a horizontal spiral of the form (2.12). We work in exponential coordinates of the second type centered at γ(0).
We fix on D the metric g making orthonormal the vector fields X 1 and X 2 spanning D. This is without loss of generality, because any other metric is equivalent to this one in a neighborhood of the center of the spiral. With this choice, the length of γ is the standard length of its horizontal coordinates and for a spiral as in (2.12) we have L(γ) = In particular, γ is rectifiable precisely when tφ ∈ L 1 (0, 1), and κ is a Lipschitz curve in the plane precisely when tφ ∈ L ∞ (0, 1). For k ∈ N andĒ = (E 3 , . . . , E n ), we denote by D(γ; k,Ē ) the curve constructed in Section 3. The devices E j = (h j , η j , ε j ) are chosen in such a way that the parameters h j , η j are fixed as in Theorem 5.1 and ε 3 , . . . , ε n are the unique solutions to the system (5.2), for k large enough. In this way the curves γ and D(γ; k,Ē )(1) have the same initial and end-point.
We claim that for k ∈ N large enough the length of D(γ; k,Ē ) is less than the length of γ. We denote by ∆L(k) = L(D(γ; k,Ē )) − L(γ) the gain of length and, namely, |ε j |.
We used (4.5) and the fact that w i ≥ 2. The new constants C 2 , C 3 do not depend on k. By (6.2) and (6.3), the inequality ∆L(k) > 0 is implied by where C 4 is a large constant independent of k. For any k ∈ N, we split the interval F k = F + k ∪ F − k where F + k = {t ∈ F k : |tφ(t)| ≥ 1} and F − k = {t ∈ F k : |tφ(t)| < 1}.
On the set F + k we have where the last inequality holds for all k ∈ N large enough, and namely as soon as 3C 4 t k < 1. On the set F − k we have where the last inequality holds for all k ∈ N large enough, by our assumption on the spiral lim t→0 + |φ(t)| = ∞. Now (6.5) and (6.6) imply (6.4) and thus ∆L(k) > 0. This ends the proof of Theorem 1.1.