Non-minimality of spirals in sub-Riemannian manifolds

https://doi.org/10.1007/s00526-021-02077-4

Calculus of Variations

Non-minimality of spirals in sub-Riemannian manifolds

Roberto Monti¹ ·Alessandro Socionovo¹

Received: 26 January 2021 / Accepted: 2 August 2021 / Published online: 4 September 2021

© The Author(s) 2021

Abstract

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.

Mathematics Subject Classification 53C17·49K30·28A75·49K21

1 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 sur- jective). There exist singular curves that are as a matter of fact length-minimizing. The first example was discovered in [12] and other classes of examples (regular abnormal extremals) are studied in [11]. 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 [10]

and [4] it was proved that length minimizers cannot have singular points of this kind. These results have been improved in [14]: at any point, the tangent cone to a length-minimizing curve contains at least one line (a half line, for extreme points), see also [5]. The uniqueness

Communicated by A. Malchiodi.

B

1 Dipartimento di Matematica “Tullio Levi-Civita”, Università di Padova, via Trieste 63, 35121 Padua, Italy

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 Example2.6below. These curves may appear as Goh extremals in Carnot groups, see [8] and [9, Section 5]. For these reasons, the results of [14] 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.

LetM be ann-dimensional,n ≥3, analytic manifold endowed with a rank 2 analytic distribution D ⊂ T M that is bracket generating (Hörmander condition). An absolutely continuous curveγ ∈ AC([0,1];M) is horizontal ifγ˙ ∈ D(γ )almost everywhere. The length ofγ is defined fixing a metric tensorgonDand letting

L(γ )=

[0,1]g_γ(γ ,˙ γ )˙ ^1/2dt. (1.1) The curveγ is a length-minimizer between its end-points if for any other horizontal curve

¯

γ ∈AC([0,1];M)such thatγ (¯ 0)=γ (0)andγ (¯ 1)=γ (1)we haveL(γ )≤L(γ )¯ . Our notion of horizontal spiral in a sub-Riemannian manifold of rank 2 is fixed in Definition 1.1. We will show that spirals are not length-minimizing when the horizontal distributionD satisfies the following commutativity condition. Fix two vector fields X1,X2 ∈Dthat are linearly independent at some pointp∈M. Fork ∈Nand for a multi-indexJ =(j₁, . . . ,j_k), with j_i ∈ {1,2}, we denote byX_J = [X_j1,[. . . ,[X_jk−1,X_jk] · · · ]]the iterated commutator associated withJ. We define its length as the length of the multi-index J, i.e., len(X_J)= len(J)=k. Then, our commutativity assumption is that, in a neighborhood of the pointp,

[XI,XJ] =0 for all multi-indices with len(I),len(J)≥2. (1.2) This condition is not intrinsic and depends on the basisX₁,X₂of the distributionD. After introducing exponential coordinates of the second type, the vector fields X₁,X₂ can be assumed to be of the form (2.3) below, and the pointpwill be the center of the spiral.

A curve γ ∈ AC([0,1];M) is horizontal if γ (t)˙ ∈ D(γ (t))for a.e. t ∈ [0,1]. In coordinates we haveγ =(γ1, . . . , γn)and, by (2.3), theγj’s satisfy for j =3, . . . ,nthe following integral identities

γj(t)=γj(0)+ _t

0

aj(γ (s))γ˙2(s)ds, t∈ [0,1]. (1.3) Whenγ (0)andγ1, γ2are given, these formulas determine in a unique way the whole hor- izontal curveγ. We callκ ∈ AC([0,1];R²),κ = (γ1, γ2), the horizontal coordinates of γ.

Definition 1.1 (Spiral) We say that a horizontal curveγ ∈ AC([0,1];M)is a spiralif, in exponential coordinates of the second type centered atγ (0), the horizontal coordinates κ∈AC([0,1];R²)are of the form

κ(t)=te^iϕ(t), t∈]0,1], (1.4)

whereϕ ∈C¹(]0,1];R⁺)is a function, calledphaseof the spiral, such that|ϕ(t)| → ∞ and| ˙ϕ(t)| → ∞ast →0⁺. The pointγ (0)is called center of the spiral.

A priori, Definition1.1depends on the basisX1,X2ofD, see however our comments about its intrinsic nature in Remark2.5. Without loss of generality, we shall focus our attention on spirals that are oriented clock-wise, i.e., with a phase satisfyingϕ(t)→ ∞andϕ(t)˙ → −∞

ast→0⁺. Such a phase is decreasing near 0. Notice that ifϕ(t)→ ∞andϕ(˙ t)has a limit ast→0⁺then this limit must be−∞.

Our main result is the following

Theorem 1.2 Let(M,D,g)be an analytic sub-Riemmanian manifold of rank 2 satisfying (1.2). Any horizontal spiralγ ∈ AC([0,1];M)is not length-minimizing near its center.

Differently from [4,5,10,14] and similarly to [13], 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 Theorem1.2is in the deeper understanding that it provides on the loss of minimality caused by singularities.

The proof of Theorem1.2consists 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 Theorem2.2, that may be of independent interest.

In Sect.3, we start the construction of the competing curve. Here we use the specific structure of a spiral. The curve obtained by cutting one spire near the center is shorter. The error appearing at the end-point will be corrected modifying the spiral in a certain number of locations with “devices” depending on a set of parameters. The horizontal coordinates of the spiral are a planar curve intersecting the positivex₁-axis infinitely many times. The possibility of adding devices at such locations arbitrarily close to the origin will be a crucial fact.

In Sect.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 Sect.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 Theorem5.1by a linearization argument.

Sections3–5contain the technical core of the paper.

We use the specific structure of the length-functional in Sect.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 Theorem1.2.

We briefly comment on the assumptions made in Theorem1.2. The analyticity ofMand Dis needed only in Sect.2. In the analytic case, it is known that length-minimizers are smooth in an open and dense set, see [15]. See also [3] for aC¹-regularity result whenMis an analytic manifold of dimension 3.

The assumption that the distributionDhas 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 end-point seems extremely difficult when the coefficients of the horizontal vector fields depend also on non- horizontal coordinates, see Remark4.3.

We thank Marco F. Sica for his help with the pictures and the referee for his comments that improved the exposition of the paper.

2 Exponential coordinates at the center of the spiral

In this section, we introduce inMexponential coordinates of the second type centered at a pointp∈M, that will be the center of the spiral.

LetX1,X2 ∈Dbe linearly independent atp. Since the distributionDis bracket-generating we can find vector-fields X3, . . . ,Xn, withn = dim(M), such that each Xi is an iterated commutator ofX1,X2with lengthwi =len(Xi),i =3, . . . ,n, and such thatX1, . . . ,Xn

atpare a basis forT_pM. By continuity, there exists an open neighborhoodUofpsuch that X₁(q), . . . ,X_n(q)form a basis forT_qM, for anyq ∈ U. We callX₁, . . . ,X_n a stratified basis of vector-fields inM.

Letϕ∈C^∞(U;Rⁿ)be a chart such thatϕ(p)=0 andϕ(U)=V, withV ⊂Rⁿ open neighborhood of 0 ∈ Rⁿ. Then X1 = ϕ_∗X1, . . . ,Xn = ϕ_∗Xn is a system of point-wise linearly independent vector fields inV ⊂Rⁿ. Since our problem has a local nature, we can without loss of generality assume thatM=V=Rⁿandp=0.

After these identifications, we have a stratified basis of vector-fieldsX₁, . . . ,X_n inRⁿ. We say thatx=(x1, . . . ,xn)∈Rⁿare exponential coordinates of the second type associated with the vector fieldsX1, . . . ,Xnif we have

x=^X_x1¹◦ · · · ◦_x^X_nⁿ(0), x∈Rⁿ. (2.1) We are using the notations^X = exp(s X),s ∈R, to denote the flow of a vector-fieldX.

From now on, we assume without loss of generality thatX₁, . . . ,X_nare complete and induce exponential coordinates of the second type.

We define the homogeneous degree of the coordinatexi ofRⁿ aswi = len(Xi). We introduce the 1-parameter group of dilationsδ_λ:Rⁿ →Rⁿ,λ >0,

δλ(x)=(λ^w1x1, . . . , λ^wnxn), x ∈Rⁿ,

and we say that a function f :Rⁿ →Risδ-homogeneous of degreew∈Nif f(δλ(x))= λ^wf(x)for allx ∈Rⁿ andλ >0. An example ofδ-homogeneous function of degree 1 is the pseudo-norm

x = n

j=1

|xi|^1/wi, x∈Rⁿ. (2.2)

The following theorem is proved in [6] in the case of general rank. A more modern approach to nilpotentization can be found in [2] and [7].

Theorem 2.1 LetD=span{X1,X2} ⊂T M be an analytic distribution of rank 2. In expo- nential coordinates of the second type around a point p∈ M identified with0 ∈Rⁿ, the vector fields X₁and X₂have the form

X₁(x)=∂x1, X2(x)=∂x2+

n j=3

aj(x)∂xj, (2.3)

for x ∈ U , where U is a neighborhood of0. The analytic functions aj ∈ C^∞(U), j = 3, . . . ,n, have the structure a_j =p_j+r_j, where:

(i) p_jareδ-homogeneous polynomials of degreewj−1such that p_j(0,x₂, . . . ,x_n)=0;

(ii) r_j ∈C^∞(U)are analytic functions such that, for some constants C₁,C₂ >0and for x∈U ,

|rj(x)| ≤C1x^wj and |∂xirj(x)| ≤C2x^wj−wi. (2.4) Proof The proof thataj= pj+rjwherepjare polynomials as in (i) and the remaindersrj

are real-analytic functions such thatr_j(0)=0 can be found in [6]. The proof of (ii) is also implicitly contained in [6]. Here, we add some details. The Taylor series ofr_jhas the form

rj(x)= ∞ =wj

r_j (x)= ∞ =wj

α∈A

c_α x^α,

where_A = {α ∈ Nⁿ : α1w1 + · · · +αnwn = },x^α = x₁^α1· · ·x_n^αn andc_α ∈ Rare constants. Here and in the following,N= {0,1,2, . . .}. The series converges absolutely in a small homogeneous cubeQ_δ= {x ∈Rⁿ : x ≤δ}for someδ >0, and in particular

∞ =wj

δ

α∈A

|c_α |<∞.

Using the inequality|x^α| ≤ x forα∈A, forx∈Q_δwe get

|rj(x)| ≤C1x^wj, withC1= ∞ =wj

δ ^−wj

α∈A

|c_α|<∞.

The estimate for the derivatives ofr_jis analogous. Indeed, we have

∂xirj(x)= ∞ =wj

α∈A

αic_α x^α−ei,

whereα−ei ∈A −wi wheneverα∈A. Above, ei =(0, . . . ,1, . . . ,0)with 1 at positioni is the canonicalith versor ofRⁿ. Thus the leading term in the series has homogeneous degree wj−wi and repeating the argument above we get the estimate|∂x_irj(x)| ≤C2x^wj−wi

forx ∈Q_δ.

When the distribution_Dsatisfies the commutativity assumption (1.2) the coefficientsa_j appearing in the vector-fieldX2in (2.3) enjoy additional properties. In the next theorem, the specific structure of exponential coordinates of the second type will be very helpful in the computation of various derivatives. In particular, in Lemma2.3we need a nontrivial formula from [6, Appendix A], given in such coordinates.

Theorem 2.2 If_D ⊂ T M is an analytic distribution of rank 2 satisfying (1.2)then the functions a3, . . . ,anof Theorem2.1depend only on the variables x1and x2.

Proof Let:R×Rⁿ→Rⁿbe the map(t,x)=_t^X2(x),wherex∈Rⁿandt ∈R. Here, we are using the exponential coordinates (2.1). In the following we omit the composition sign

◦. Defining:R³×Rⁿ →Rⁿ as the mapt,x₁,x₂(p)=_−(x^X2_2+t)−x^X1₁^X_t²x^X1¹x^X2²(p), we have

(t,x)=x^X₁¹x^X₂²+tt,x1,x2x^X₃³. . . ^Xx_nⁿ(0).

We claim that there exists aC >0 independent oftsuch that, fort→0,

|_t,x1,x2s^Xj −s^Xj_t,x1,x2| ≤Ct². (2.5)

We will prove claim (2.5) in Lemma2.3below. From (2.5) it follows that there exist mappings R_t∈C^∞(Rⁿ,Rⁿ)such that

(t,x)=_x^X₁^1X_x2²_+tx^X₃³. . . ^X_xnⁿ_t,x1,x2(0)+R_t(x), (2.6) and such that|Rt| ≤Ct²fort →0.

By the structure (2.3) of the vector fieldsX₁andX₂and sincet,x1,x2is the composition ofC^∞maps, there existC^∞functions f_j = f_j(t,x₁,x₂)such that

t,x1,x2(0)=

0,0,f₃(t,x₁,x₂), . . . ,f_n(t,x₁,x₂)

=exp ⁿ

j=3

f_j(t,x₁x₂)X_j (0).

(2.7) By (1.2), from (2.6) and (2.7) we obtain

(t,x)=^Xx₁^1Xx2²+texp ⁿ

i=3

(x_j+ f_j(t,x₁,x₂))X_j

(0)+R_t(x)

=

x₁,x₂+t,x₃+ f₃(t,x₁,x₂), . . . ,x_n+ f_n(t,x₁,x₂)

+R_t(x), and we conclude that

X2(x)= d

dt(x,t)

t=0 =∂2+ n

j=3

d

dt fj(t,x1,x2)

t=0∂j.

Thus the coefficientsa_j(x1,x₂) = _dt^d f_j(t,x₁,x₂)|_t=0, j = 3, . . . ,n, depend only on the

first two variables, completing the proof.

In the following lemma, we prove our claim (2.5).

Lemma 2.3 LetD ⊂ T M be an analytic distribution satisfying(1.2). Then for any j = 3, . . . ,n the claim in(2.5)holds.

Proof Let X = Xj for any j = 3, . . . ,n and define the mapT_t,x^X

1,x2;s = t,x₁,x_2s^X− _s^Xt,x1,x2. Fort=0 the map0,x1,x2is the identity and thusT_0,x^X

1,x2;s =0. So, claim (2.5) follows as soon as we show that

T˙_0,x^X

1,x2;s = ∂

∂t

t=0T_t,x^X

1,x2;s=0, for anys∈Rand for allx₁,x₂∈R.

We first compute the derivative oft,x₁,x₂with respect tot. Lettingt,x₁ =_−x^X1_1t^X2Xx1¹

we havet,x₁,x₂ = _−(x^X2 _2+t)t,x₁x^X₂²,and, thanks to [6, Appendix A], the derivative of _t,x1att=0 is

˙0,x1 = ∞ ν=0

c_ν,x1W_ν,

whereW_ν = [X₁,[· · ·,[X₁,X₂] · · · ]]withX₁appearingνtimes andc_ν,x1 =(−1)^νx₁^ν/ν!. In particular, we havec_0,x1=1. Then the derivative oft,x1,x2att=0 is

˙0,x1,x2 = −X₂+d^X₋²x₂

˙0,x1(^X_x2²)

= −X₂+ ∞ ν=0

c_ν,x1d^X₋²x₂

W_ν(_x^X₂²)

=^∞

ν=1

c_ν,x1d_−x^X2₂

W_ν(_x^X₂²) , because the term in the sum withν =0 isd^X₋²x₂

X₂(^Xx2²)

= X₂. Inserting this formula for˙0,x1,x2into

T˙_0,x^X

1,x2;s = ˙0,x1,x2(_s^X)−d_s^X(˙0,x1,x2), (2.8) we obtain

T˙₀^X_,x

1,x₂;s=^∞

ν=1

c_ν,x₁d_−x^X2₂

W_ν(^X_x2²_s^X)

−d_s^X∞

ν=1

c_ν,x₁d_−x^X2₂ W_ν

_x^X₂²)

=d_s^X ∞ ν=1

c_ν,x1

d_−s^X d_−x^X2₂

W_ν(_x^X₂²_s^X)

−d_−x^X2₂

W_ν(_x^X₂²) . In order to prove thatT˙_0,x^X

1,x2;svanishes for allx₁,x₂ands, we have to show that g(x₂,s):=d₋^Xsd_−x^X2₂

W_ν(x^X₂²s^X)

−d_−x^X2₂

W_ν(x^X₂²)

=0, (2.9)

for anyν≥1 and for anyx₂ands. From₀^X =id it follows thatg(x₂,0)=0. Then, our claim (2.9) is implied by

h(x2,s):= ∂

∂sg(x2,s)=0. (2.10)

Actually, this is a Lie derivative and, namely, h(x₂,s)= −d₋^Xs

X,d^X₋²x₂

W_ν(^Xx₂²) .

Notice thath(0,s)= −d₋^Xs[X,W_ν] =0 by our assumption (1.2). In a similar way, for any k∈Nwe have

∂^k

∂x₂^kh(0,s)=(−1)^k+1d₋^Xs[X,[X₂,· · · [X₂,W_ν] · · · ]] =0,

with X2 appearingk times. Since the functionx2 → h(x2,s)is analytic our claim (2.10)

follows.

We conclude this sections with some general remarks.

Remark 2.4 By Theorem2.2, we can assume thataj(x) =aj(x1,x2)are functions of the variablesx1,x2. In this case, formula (1.3) for the coordinates of a horizontal curveγ ∈ AC([0,1];M)reads, for j=3, . . . ,n,

γj(t)=γj(0)+ _t

0

a_j(γ1(s), γ2(s))γ˙2(s)ds, t∈ [0,1]. (2.11)

Remark 2.5 Definition1.1of horizontal spiral is stable with respect to change of coordinates in the following sense.

After fixing exponential coordinates, we have that 0 ∈ Rⁿ is the center of the spiral γ : [0,1] →Rⁿ, with horizontal projectionκ(t)as in Definition1.1.

We consider a diffeomorphismF∈C^∞(Rⁿ;Rⁿ)such thatF(0) =0. In the new coor- dinates, our spiralγ isζ(t) = F(γ (t )). We define the horizontal coordinates ofζ in the following way: the setd₀F(D(0)), whered₀Fis the differential ofFat 0, is a 2-dimensional subspace ofRⁿ =Im(d₀F); denoting byπ :Rⁿ →d₀F(D(0))the orthogonal projection, we define the horizontal coordinates ofζasξ(t)=π(F(γ (t)).

We claim thatξ, in the planed0F(D(0)), is of the form (1.4), with a phaseωsatisfying

|ω| → ∞and| ˙ω| → ∞. In particular, these properties ofξare stable up to isometries of the plane. Then, we can assume thatξ(t) =(F₁(γ (t)),F₂(γ (t))), withF_i :Rⁿ →Rof class C^∞, fori =1,2. In this setting, we will show that| ˙ω| → ∞.

The functions(t)= |ξ(t)| = |(F1(γ (t)),F2(γ (t)))|satisfies

0<c₀ ≤ ˙s(t)≤c₁<∞, t∈(0,1]. (2.12) Define the functionω ∈ C¹((0,1])letting ξ(t) = s(t)e^iω(s(t)). Then differentiating the identity obtained inverting

tan ω(s(t))

= F₂(γ (t))

F₁(γ (t)), t∈(0,1], we obtain

˙

s(t)ω(s(t))˙ = 1

s(t)²(γ (t)),γ (t),˙ t∈(0,1], (2.13) where the function(x) = F1(x)∇F2(x)−F2(x)∇F1(x)has the Taylor development as x →0

(x)=∇F1(0),x∇F2(0)− ∇F2(0),x∇F1(0)+O(|x|²).

Observe that from (2.11) it follows that| ˙γj(t)| =O(t)forj≥3. Denoting by ¯∇the gradient in the first two variables, we deduce that ast→0⁺we have

(γ ),γ˙ = F1(γ )¯∇F2(γ )−F2(γ )¯∇F1(γ ),κ +˙ O(t²) (2.14) with

F1(γ )¯∇F2(γ )−F2(γ )¯∇F1(γ )= ¯∇F1(0), κ ¯∇F2(0)− ¯∇F2(0), κ ¯∇F1(0)+O(t²).

Inserting the last identity andκ˙ = e^iϕ +i tϕe˙ ^iϕ into (2.14), after some computations we obtain

(γ ),γ˙ = ˙ϕt²det(d0F(0))¯ +O(t²),

where det(d₀F¯(0)) = 0 is the determinant Jacobian at x₁ = x₂ = 0 of the mapping (x₁,x₂) →(F₁(x₁,x₂,0),F₂(x₁,x₂,0)). Now the claim| ˙ω(s)| → ∞ass → 0⁺easily follows from (2.12), (2.13) and from| ˙ϕ(t)| → ∞ast→0⁺.

Example 2.6 An interesting example of horizontal spiral is the double-logarithm spiral, the horizontal lift of the curveκin the plane of the form (1.4) with phaseϕ(t)=log(−logt), t∈(0,1/2]. In this case, we have

˙

ϕ(t)= 1

tlogt, t∈(0,1/2],

and clearlyϕ(t)→ ∞andϕ(˙ t)→ −∞ast→0⁺. In fact, we also havetϕ˙∈L^∞(0,1/2), which means thatκand thusγis Lipschitz continuous. This spiral has the following additional properties:

(i) for any v ∈ R² with |v| = 1 there exists an infinitesimal sequence of positive real numbers(λn)n∈Nsuch thatκ(λnt)/λn→tvlocally uniformly, asn→ ∞;

(ii) for any infinitesimal sequence of positive real numbers(λn)_n∈Nthere exists a subsequence and av∈R²with|v| =1 such thatκ(λnkt)/λnk→tvask → ∞, locally uniformly.

This means that the tangent cone ofκatt=0 consists of all half-lines inR²emanating from 0.

3 Cut and correction devices

In this section, we begin the construction of the competing curve. Letγ be a spiral with horizontal coordinatesκas in (1.4). We can assume thatϕis decreasing and thatϕ(1)=1 and we denote byψ : [1,∞)→(0,1]the inverse function ofϕ. Fork∈Nandη∈ [0,2π) we definet_kη∈(0,1]as the unique solution to the equationϕ(t_kη)=2πk+η, i.e., we let t_kη=ψ(2πk+η). The times

tk=tk0=ψ(2πk), k∈N, (3.1)

will play a special role in our construction. The pointsκ(tk)are in the positivex1-axis.

For a fixedk ∈N, we cut the curveκin the interval[t_k+1,t_k]following the line segment joiningκ(t_k+1)toκ(t_k)instead of the pathκ, while we leave unchanged the remaining part of the path. We call this new curveκ_k^cutand, namely, we let

κ_k^cut(t)=κ(t) for t∈ [0,t_k+1] ∪ [t_k,1], κ_k^cut(t)=(t,0) for t∈ [tk+1,tk].

We denote byγ_k^cut∈AC([0,1];M)the horizontal curve with horizontal coordinatesκ_k^cut and such thatγ_k^cut(0)=γ (0). Fort∈ [0,t_k+1], we haveγ_k^cut(t)=γ (t). To correct the errors produced by the cut on the end-point, we modify the curveκ_k^cut using a certain number of devices. The construction is made by induction.

We start with the base construction. LetE = (h, η, ε)be a triple such that h ∈ N, 0 < η < π/4, andε ∈ R. Starting from a curve κ : [0,1] → R², we define the curve D(κ;E): [0,1+2|ε|] →R²in the following way:

D(κ;E)(t)=

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎩

κ(t) t∈ [0,t_hη]

κ(thη)+(sgn(ε)(t−thη),0) t∈ [thη,thη+ |ε|]

κ(t− |ε|)+(ε,0) t∈ [thη+ |ε|,th+ |ε|]

κ(t_h)+(2ε+sgn(ε)(t_h−t),0)t∈ [t_h+ |ε|,t_h+2|ε|]

κ(t−2|ε|) t∈ [t_h+2|ε|,1+2|ε|].

(3.2)

We denote by D(γ;E)the horizontal curve with horizontal coordinates D(κ;E). We let D(γ˙ ;E)= _dt^dD(γ;E)and we indicate by Di(γ;E)the i-th coordinate of the corrected curve in exponential coordinates.

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, whereD˙₂(γ;E) =0,

and then reparameterize the curve on[0,1]. Namely, we define the discontinuous curve D(κ;E): [0,1] →R²as

D(κ;E)(t)=

⎧⎨

⎩

κ(t) t∈ [0,thη] κ(t)+(ε,0)t∈(thη,th) κ(t) t∈ [t_h,1].

(3.3)

The “formal”ith coordinate of the curve D(κ;E)is given by Di(γ;E)(t)=

_t

0

ai(D(κ;E)(s))˙κ2(s)ds.

The following identities withε >0 can be checked by an elementary computation

D(γ;E)(t)=

⎧⎨

⎩

D(γ;E)(t) t∈ [0,t_hη] D(γ;E)(t+ε) t∈(t_hη,t_h) D(γ;E)(t+2ε)t∈ [t_h,1].

(3.4)

Whenε <0 there are similar identities. With this notation, the final error produced on the ith coordinate by the correction deviceE is

γi(1)−D_i(γ;E)(1+2|ε|)= ₁

0

a_i(κ(s))−a_i(D(κ;E)(s))

˙

κ2(s)ds. (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 triplesE. We first fix the number of triples and iterations.

Fori = 3, . . . ,n, letBi = {(α, β) ∈ N² : α+β = wi −2}, wherewi ≥ 2 is the homogeneous degree of the coordinatex_i. Then, the polynomials p_i given by Theorem2.1 and Theorem2.2are of the form

pi(x1,x2)=

(α,β)∈Bi

c_αβx₁^α+1x₂^β, (3.6)

for suitable constantsc_αβ ∈R. We set

= n i=3

Card(Bi), (3.7)

and we consider an( −2)-tuple of triples_E¯=(E3, . . . ,E)such thath <h ₋₁<· · ·<

h3<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 thatc_αβ=0 for all(α, β)∈Bi but one. Namely, we can assume that

pi(x1,x2)=x₁^αi+1x₂^βi with αi+βi =wi−2, (3.8) and withc_αiβi =1. In this case, we have =nand we will usen−2 devices associated with the triples_E3, . . . ,Ento correct the coordinatesi=3, . . . ,n. By the bracket generating property of the vector fieldsX1andX2and by the stratified basis property forX1, . . . ,Xn, the pairs(αi, βi)satisfy the following condition

(αi, βi)=(αj, βj) for i= j. (3.9) In the general case (3.6), we use a larger number ≥nof devices, one for each monomial x₁^α+1x₂^β appearing in p3(x1,x2), . . . ,pn(x1,x2), and we correct the error produced by the cut on each monomial. The argument showing the nonminimality of the spiral will be the same. So, from now on in the rest of the paper we will assume that the polynomialsp_iare of the form (3.8) with (3.9).

Now we clarify the inductive step of our construction. Let_E3 =(h₃, η3, ε3)be a triple such thath3 <k. We define the curveκ⁽³⁾=D(κ_k^cut;E3). Given a tripleE4 =(h4, η4, ε4) withh₄ < h₃ we then defineκ⁽⁴⁾ = D(κ⁽³⁾;E4). By induction on ∈ N, given a triple E =(h , η , ε )withh <h ₋₁, we defineκ^{( )}=D(κ^{( −1)};E). When =nwe stop.

We define the planar curve D(κ;k,_E¯) ∈ AC([0,1+2¯ε];R²) as D(κ;k,_E¯) = κ⁽ⁿ⁾ according to the inductive construction explained above, whereε¯= |ε3|+· · ·+|εn|. Then we call D(γ;k,_E¯)∈AC([0,1+2ε];¯ M), the horizontal lift of D(κ;k,_E¯)with D(γ;k,E)(0)= γ (0), the modified curve ofγ associated with_E¯and with cut of parameterk∈N. There is a last adjustment to do. In[0,1+2ε]¯ there are 2(n−2)subintervals whereκ˙₂⁽ⁿ⁾ =0. On each of these intervals the coordinates Dj(γ;k,_E¯)are constant. According to the procedure explained in (3.2)–(3.4), we erase these intervals and we parametrize the resulting curve on [0,1]. We denote this curve byγ¯=D(γ;k,_E¯).

Definition 3.1 (Adjusted modification ofγ)We call the curveγ¯=D(γ;k,_E¯): [0,1] →M the adjusted modification ofγ relative to the collections of devices_E¯ =(E3, . . . ,En)and with cut of parameterk.

Our next task is to compute the error produced by cut and devices on the end-point of the spiral. Fori =3, . . . ,nand fort∈ [0,1]we let

^γ_i(t)=a_i(κ(t))κ˙2(t)−a_i(κ(¯ t))˙¯κ2(t). (3.10) Whent<t_k+1ort>t_kwe haveκ˙2= ˙¯κ2and so the definition above reads

^γ_i(t)=

a_i(κ(t))−a_i(κ(¯ t))

˙ κ2(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:

E_i^k,E¯=γi(¯t)− ¯γi(¯t)= _t¯

t_k+1

^γ_i(t)dt

=

F_k^γ_i(t)dt+ n

j=3 A_j^γ_i(t)dt+

B_j^γ_i(t)dt .

(3.11)

In (3.11) and in the following, we use the following notation for the intervals:

Fk= [tk+1,tk], Aj = [th_j−1,th_jηj], Bj = [th_jηj,th_j], (3.12) withth₂=tk. We used also the fact that on[0,tk+1]we haveγ = ¯γ.

On the intervalF_kwe haveκ˙¯2=0 and thus

Fk

^γ_i dt =

Fk

pi(κ)+ri(κ)

˙

κ2dt. (3.13)

On the intervalsA_jwe haveκ= ¯κand thus

Aj

^γ_idt =0, (3.14)

because the functionsaidepend only onκ. Finally, on the intervalsBjwe haveκ¯1=κ1+εj

andκ2= ¯κ2and thus

Bj

_i^γdt=

Bj

{pi(κ)−pi(κ+(εj,0))}˙κ2dt+

Bj

{ri(κ)−ri(κ+(εj,0))}˙κ2dt.

(3.15) Our goal is to findk∈Nand devices_E¯such thatE_i^k,E¯=0 for alli =3, . . . ,nand such that the modified curve D(γ;k,_E¯)is shorter thanγ.

4 Effect of cut and devices on monomials and remainders

Letγ be a horizontal spiral with horizontal coordinatesκ ∈ AC([0,1];R²) of the form (1.4). 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).

Forα, β ∈N, we associate with the monomial p_αβ(x₁,x₂) =x^α+₁ ¹x₂^βthe functionγαβ

defined fort∈ [0,1]by γ_αβ(t)=

κ|[0,t]

p_αβ(x1,x₂)d x2= _t

0

p_αβ(κ(s))κ˙2(s)ds.

Whenpi = p_αβ, the functionγ_αβis the leading term in theith coordinate ofγin exponential coordinates. In this case, the problem of estimatingγi(t)reduces to the estimate of integrals of the form

I_ωη^αβ= _tω

t_η

κ1(t)^α+1κ2(t)^βκ˙2(t)dt, (4.1) whereω≤ηare angles,t_ω=ψ(ω)andt_η=ψ(η). Forα, β ∈N,h∈Nandη∈(0, π/4) we also let

j_hη^αβ=η^β

_2hπ+η

2hπ t_ϑ^α+β+2dϑ= _th

t_hη

t^α+β+2| ˙ϕ(t)|dt, (4.2) where in the second equality we setϑ =ϕ(t).

Proposition 4.1 There exist constants0<c_αβ <C_αβdepending onα, β∈Nsuch that for all h∈Nandη∈(0, π/4)we have

c_αβj_h^αβ_η ≤ |I_2h^αβ_π,2hπ+η| ≤C_αβj_h^αβ_η. (4.3) Before proving this proposition, we notice that the integralsI_ωη^αβin (4.1) are related to the integrals

J_ωη^αβ= _η

ω t_ϑ^α+β+2cos^α(ϑ)sin^β(ϑ)dϑ. (4.4) Lemma 4.2 For anyα, β∈Nandω≤ηwe have the identity

(α+β+2)I_ωη^αβ=t_ω^α+β+2D^αβ_ω −t_η^α+β+2D^αβ_η −(α+1)J_ωη^αβ, (4.5) where we set D^αβ_ω =cos^α+1(ω)sin^β+1(ω).

Proof Inserting intoI_ωη^αβthe identitiesκ1(t)=tcos(ϕ(t)),κ2(t)=tsin(ϕ(t)), andκ˙2(t)= sin(ϕ(t))+tcos(ϕ(t))ϕ(t)˙ we get

I_ωη^αβ = _tω

t_η

t^α+β+1D_ϕ(^αβ_t)dt+ _tω

t_η

t^α+β+2cos^α+2(ϕ(t))sin^β(ϕ(t))ϕ(˙ t)dt, and, integrating by parts in the first integral, this identity reads

I_ωη^αβ=

t^α+β+2D^αβ_ϕ(t) α+β+2

t_ω t_η

+ α+1 α+β+2

_tω

t_η

t^α+β+2cos^α(ϕ(t))sin^β+2(ϕ(t))ϕ(t˙ )dt

− β+1 α+β+2

_tω

t_η

t^α+β+2cos^α+2(ϕ(t))sin^β(ϕ(t))ϕ(˙ t)dt +

_tω

t_η

t^α+β+2cos^α+2(ϕ(t))sin^β(ϕ(t))ϕ(t)dt.˙

Grouping the trigonometric terms and then performing the change of variableϕ(t)=ϑ, we get