END-POINT EQUATIONS AND REGULARITY OF SUB-RIEMANNIAN GEODESICS

GIAN PAOLO LEONARDI AND ROBERTO MONTI

Abstract. For a large class of equiregular sub-Riemannian manifolds, we show that length minimizing curves have no corner-like singularities. Our first result is the reduction of the problem to the homogeneous, rank 2 case, by means of a nilpotent approximation. We also identify a suitable condition on the tangent Lie algebra implying existence of a horizontal basis of vector fields whose coefficients depend only on the first two coordinates x_{1}, x_{2}. Then, we cut the corner and lift the new curve to a horizontal one, obtaining a decrease of length as well as a perturbation of the end-point. In order to restore the end-point at a lower cost of length, we introduce a new iterative construction, which represents the main contribution of the paper. We also apply our results to some examples.

1. Introduction

Let M be an n-dimensional smooth manifold with n ≥ 3, and let D be an mdimensional, completely non integrable (bracket generating) distribution on M. A Lipschitz curve γ : [0, 1] → M is D-horizontal (or simply horizontal) if its derivative γ̇(t) lies in D(γ(t)) for a.e. t ∈ [0, 1]. If M is connected, then, by Chow-Rashevsky theorem, any pair of points in M can be connected by a horizontal curve. For a fixed positive quadratic form g on D, a distance can be introduced in M, on minimizing the length of horizontal curves connecting points. The triple (M, D, g) is called sub-Riemannian manifold. The resulting metric space is locally compact. If it is also complete, then length minimizing curves between two points always exist, by Hopf-Rinow theorem in length spaces.

Length minimizing curves are critical points (extremals) of the end-point map, and, according to Pontryagin Maximum Principle, they can be either normal or singular. The latter case does not occur in Riemannian geometry. Normal extremals (“normal geodesics”) are curves of class C^{∞} solving the equations for sub-Riemannian geodesics. Singular extremals (“abnormal geodesics”) depend on the distribution D and not on the metric g. It was believed for some time that only normal extremals could be length minimizing (see [S]). In 1994, however, Montgomery proved this to be false, on producing an example of singular length minimizing curve [M1]. In fact, manifolds with distributions of rank 2 are rich of abnormal geodesics, as shown by Liu and Sussmann in [LS].

By now, there is a large literature on sub-Riemannian geodesics, especially on abnormal ones, and it is not possible to give, here, a comprehensive account of it. We

Date: 25th October 2006.

2 GIAN PAOLO LEONARDI AND ROBERTO MONTI

only mention the second order theory for abnormal geodesics developed by Agrachev and Sarychev in [ASa], the study of sub-analyticity of the sub-Riemannian distance function due to Agrachev and Gauthier [AG], and the existence of strictly abnormal geodesics shown by Golé and Karidi [GK].

One of the main open questions is the regularity of sub-Riemannian length minimizing curves, in fact of abnormal ones. A nice introduction to the problem and an up-to-date bibliography can be found in the book [M2], where many relations to anholonomic mechanics are discussed. We refer the reader also to the book [AS], where the problem is presented from the point of view of geometric control theory. In this paper, we address the regularity issue and, precisely, we prove that in a large class of sub-Riemannian manifolds, curves with corners are not length minimizing.

Let us introduce, first, the following assumptions. We let D_{1} = D and D_{i} = [D_{1}, D_{i−1}], i.e. D_{i} is the linear span of all commutators [D_{1}, D_{i−1}]. Then we let L_{0} = {0} and L_{i} = D_{1} + ... + D_{i}, i ≥ 1. For p ∈ M, we denote by L_{i}(p) the set of all X(p) ∈ T_{p}M with X ∈ L_{i}. By the non integrability condition, for any p ∈ M there exists r ∈ ℕ such that L_{r}(p) = T_{p}M. First, we assume that for each i = 1, ..., r

dim (L_{i}(p)/L_{i−1}(p)) is constant for p ∈ M,  (1.1)

i.e. D is a equi-regular distribution. Then, we also require the following technical condition

[L_{i}, L_{j}] ⊂ L_{i+j−1},  i, j ≥ 2, i + j ≥ 5.  (1.2)

Condition (1.2) is automatically satisfied if r ≤ 4. For distributions with step r ≥ 5, however, it is generically false. Indeed, in Carnot groups (1.2) is equivalent to [D_{i}, D_{j}] = 0 for i, j ≥ 2 and i + j ≥ 5.

Our main result, Theorem 3.1, states that in a sub-Riemannian manifold (M, D, g) satisfying (1.1) and (1.2), a horizontal curve with a corner is not length minimizing. A Lipschitz curve γ : [−1, 1] → M has a corner at the point γ(0) ∈ M if the left and right derivatives of γ at t = 0 exist and are linearly independent. To this aim, we introduce a new technique for the analysis of the end-point equations. Precisely, in a relevant situation, we show that these equations can be solved by an iterative method, which is the main contribution of this work.

By the equi-regularity assumption (1.1), we can consider the nilpotent approximation of (M, D). We blow up the manifold M, the distribution D and the curve γ at the corner. In the limit, length minimality is preserved. Hence, we can assume M = ℝ^{n} and that γ : [−1, 1] → ℝ^{n} is

γ(t) = \begin{cases}(0, −t, 0, ..., 0) & t ∈ [−1, 0], \\ (t, 0, ..., 0) & t ∈ [0, 1]. \end{cases}  (1.3)

The limit distribution is homogeneous (i.e. it admits a one-parameter family of dilations) and, in fact, we get a Carnot group, by Mitchell-Margulis-Mostow theorem. Moreover, the limit distribution can be assumed to be of rank 2 and, using exponential coordinates, it can be consequently identified with the linear span of the vector fields in ℝ^{n}

X_{1} = ∂_{1}, X_{2} = ∂_{2} + ∑_{i=3}^{n} P_{i}(x)∂_{i},  (1.4)

where P_{i} are homogeneous polynomials with respect to the given family of dilations. For homogeneous distributions assumption (1.2) implies, after a suitable choice of coordinates, that the polynomials P_{i}(x) in (1.4) depend only on coordinates x_{1} and x_{2} (see Proposition 2.6). The reduction to the homogeneous case of rank 2, as well as to variables x_{1} and x_{2}, is explained in Section 2 and will mainly help us to set up the iterative solution of the non-linear system of end-point equations (see Section 3). This reduction seems to be interesting in its own right.

With this special form of the polynomials, a detailed analysis of the end-point equations at the curve γ in (1.3) can be carried out. In order to show that γ is not length minimizing, we cut the corner by a triangle T^{η} = {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}, x_{2} > 0, x_{1} + x_{2} < η} getting a curve that follows the line segment from (0, η) to (η, 0) in the x_{1}x_{2} plane, where η > 0 is a small parameter. This curve must be lifted to get a horizontal curve. The length of the new curve is shorter of order η than the length of γ. However, the lifting perturbs the end-point, and precisely, the error appearing in the coordinate i ≥ 3 is

T_{i}^{η} = ∫_{∂T^{η}} P_{i}(x_{1}, x_{2}) dx_{2}.

In Theorem 3.12, we show that the end-point can be restored paying a cost of length of order η^{4/3−ε}, ε > 0 (see (3.61)). The end-point of the lifted curve is corrected by deviating the curve along the boundary of rectangles depending on various parameters: size parameters ε_{i}, position parameters b_{i}, and induction parameters λ_{i}. We need four different types of such rectangles (see (3.4)). The end-point equations form a non-linear system of n − 2 equations in the variables ε_{i} and with data T_{i}^{η}. The linearization of the system is typically singular. In order to get a solution satisfying the η^{4/3−ε} estimate, we use an iterative method which can be summarized as follows.

i) The set P of monomials appearing in the polynomials P_{i} in (1.4) is partitioned into suitable subsets P_{k} and Q_{k}. In order to correct the whole error T_{i}^{η} associated with P_{i}, we begin by correcting the error associated with each P_{k} and Q_{k}. Our approach is explained in detail at the beginning of Section 3.

ii) The partitioning scheme is chosen in such a way to have a good step by step linearization. Precisely, for each P_{k} and Q_{k} we have a system of equations with linearization of Vandermonde type. At this stage, we fix position parameters b_{i} of the rectangles. The outcome of our construction is a family of correction maps which describe the propagation of errors produced by solving the system of end-point equations associated with each P_{k} and Q_{k} (see Propositions 3.5 and 3.6).

iii) Composing and iterating countably many times the correction maps, we correct simultaneously the error associated with all P_{k} and Q_{k}. This is the most difficult

4 GIAN PAOLO LEONARDI AND ROBERTO MONTI

part of the work and it requires a suitable choice of induction parameters λ_{i}. This choice is made in Lemma 3.7 and is suggested by the proof of Theorem 3.11. The propagation of errors is controlled by growth parameters (see (3.50)) which finally yield the η^{4/3−ε} estimate (see (3.35) and (3.50)). The proof is by induction and is contained in Theorem 3.11, the core of the paper.

Our results are not sufficient to prove the full C^{1} regularity for length minimizing curves in sub-Riemannian manifolds (satisfying (1.1) and (1.2)). The main problem is the fact that Lipschitz curves may have singularities which blow up to lines. Consider, for instance, De Giorgi’s double-logarithmic spiral κ : [−1/2, 1/2] → ℝ^{2} = ℂ

κ(t) = te^{i log(− log |t|)},

where κ(0) = 0 (see e.g. [GMT]). The curve κ is rectifiable. Moreover, for any sequence (λ_{h})_{h∈ℕ} with λ_{h} → +∞ there is a subsequence (μ_{h})_{h∈ℕ} and there is ϑ ∈ [0, 2π) such that

lim_{h→+∞} μ_{h}κ(\frac{t}{μ_{h}} ) = te^{iϑ},  t ∈ ℝ.  (1.5)

Indeed, for any ϑ ∈ [0, 2π) there is a (μ_{h})_{h∈ℕ} giving (1.5). The curve κ is singular at t = 0, but any blow-up of κ is a line passing through the origin. So our argument does not apply.

In some cases, however, the structure of the singularities of abnormal extremals can be described and the corner removability provides a better regularity for sub-Riemannian geodesics. In the last section, we discuss some examples (in coordinates). We consider a distribution D in ℝ^{n} spanned by vector fields X_{1} and X_{2} as in (1.4), with P_{i}(x) = P_{i}(x_{1}, x_{2}), and with the metric g on D making X_{1} and X_{2} orthonormal. Then we introduce the function K : ℝ^{n−2} × ℝ^{2} → ℝ

K(μ, x) = ∑_{i=1}^{n−2} μ_{i}∂_{1}P_{i+2}(x).

A horizontal curve is singular (see our Definition 4.1) if and only if its horizontal projection κ onto the x_{1}x_{2}-plane is contained in the zero set K(μ, x) = 0 for some μ ≠ 0 (Proposition 4.2). On the other hand, if a length minimizing curve is not singular, then it is C^{∞} and moreover its horizontal projection κ solves the differential equation κ̈ = K(μ, κ)κ̇^{⊥}, for some μ ∈ ℝ^{n−2} and where κ^{⊥} = (−κ_{2}, κ_{1}) (Proposition 4.3).

From the corner removability, if every P_{i} is a polynomial in the variables x_{1} and x_{2}, then geodesics are curves of class C^{1} (Example 4.4). In particular, if every P_{i} is a polynomial such that ∂_{1}P_{i} has degree at most 2, then the zero sets K(μ, x) = 0 are conics and, then, all length minimizing curves are in fact smooth. Carnot groups with rank 2 and step at most 4 fall into this discussion (Example 4.6).

A brief survey of the paper is in order. In Section 2, we explain the reduction to the homogeneous, rank 2 case. The substantial part of the paper is contained in Section

END-POINT EQUATIONS 5

3. Here, we solve our system of end-point equations and we prove that curves with corners are not length minimizing. In Section 4, we discuss the examples.

Finally, it is a pleasure to thank Paolo Ciatti, who suggested to us the proof of Proposition 2.6.

2. Reduction to the homogeneous rank two case

In this section, we recall the nilpotent approximation of a sub-Riemannian manifold. We refer the reader to [B] for a more comprehensive discussion of the theory and for references to the relevant literature. Throughout this section, we assume conditions (1.1) and (1.2).

We fix a point p ∈ M and we fix a maximal family of vector fields Y_{1}, ..., Y_{m_{1}} ∈ D = L_{1}, with m_{1} = m, which are linearly independent at p. For k ∈ ℕ and for any given k-vector J = (j_{1}, ..., j_{k}) of multi-indices with 1 ≤ j_{i} ≤ m_{1} for all i = 1, ..., k, denote by Y_{J} the iterated commutator

Y_{J} = [Y_{j_{1}}, [...[Y_{j_{k−1}}, Y_{j_{k}}]...]].

We say that Y_{J} is (a commutator) of length k and we write len(Y_{J}) = k. By induction, we can choose, for any k ≥ 2, a maximal family of vector fields Y_{m_{k−1}+1}, ..., Y_{m_{k}} with length k such that Y_{1}, ..., Y_{m_{k}} are linearly independent at the point p, i.e. such that Y_{1}(p), ..., Y_{m_{k}}(p) form a basis for L_{k}(p). By complete non integrability of D, after finitely many steps the selection ends and we find a system of iterated commutators Y_{1}, ..., Y_{n} giving a basis for T_{p}M. By continuity, Y_{1}, ..., Y_{n} yield a basis for T_{q}M with q in a neighborhood of p.

In a neighborhood V of p, we consider exponential coordinates x = (x_{1}, ..., x_{n}) of the first type associated with Y_{1}, ..., Y_{n}. Using such coordinates, we identify V with a neighborhood of the origin in ℝ^{n} and p with the origin. Then, since our analysis is local, we can without loss of generality assume that M = ℝ^{n} and that

x = exp (x_{1}Y_{1} + ... + x_{n}Y_{n})(0), x = (x_{1}, ..., x_{n}) ∈ ℝ^{n}.  (2.1)

With this identification, the vector fields Y_{1}, ..., Y_{n} are now vector fields in ℝ^{n} such that Y_{i}(0) = ∂_{i}, i = 1, ..., n.

If len(Y_{i}) = w_{i} ∈ ℕ, we assign to the coordinate x_{i} of ℝ^{n} the degree w_{i}. For any λ > 0, we introduce the dilation δ_{λ} : ℝ^{n} → ℝ^{n}

δ_{λ}(x) = (λ^{w_{1}}x_{1}, ..., λ^{w_{n}}x_{n}).

We say that a function f : ℝ^{n} → ℝ is δ-homogeneous of degree w ∈ ℕ ∪ {0} if f(δ_{λ}(x)) = λ^{w}f(x). The proof of the following proposition is e.g. in [MM], on p. 306.

Proposition 2.1. Let Y_{i} = a_{i1}∂_{1} +...+a_{in}∂_{n} ∈ {Y_{1}, ..., Y_{n}} be vector field in the basis with len(Y_{i}) = w_{i}. Then, for any h = 1, ..., n, it is a_{ih} = p_{ih} + r_{ih} with a_{ih}(0) = δ_{ih} (Kronecker symbol), p_{ih} δ-homogeneous polynomial of degree w_{h} − w_{i}, and

lim_{λ→+∞} λ^{w_{h}−w_{i}}r_{ih}(δ_{1/λ}(x)) = 0.  (2.2)

6 GIAN PAOLO LEONARDI AND ROBERTO MONTI

For a smooth vector field X in ℝ^{n}, we define for λ > 0 the vector field δ_{λ}X through the relation δ_{λ}X(f)(x) = X(f ∘δ_{λ})(δ_{1/λ}(x)), x ∈ ℝ^{n}. Precisely, if X = a_{1}∂_{1}+...+a_{n}∂_{n}, then we have

δ_{λ}X(x) = ∑_{h=1}^{n} λ^{w_{h}}a_{h}(δ_{1/λ}(x))∂_{h}.

Note that for smooth vector fields X and Y, there holds

δ_{λ}[X, Y] = [δ_{λ}X, δ_{λ}Y],  λ > 0.  (2.3)

Let Y_{i} ∈ {Y_{1}, ..., Y_{n}} be an element of the basis and let p_{ih}, h = 1, ..., n, be the δ-homogeneous polynomials given by Proposition 2.1. In particular, we have p_{ii} = 1 and p_{ih} = 0 if h < i. For i = 1, ..., n define the vector fields

Y_{i}^{∞} = ∂_{i} + ∑_{h=i+1}^{n} p_{ih}∂_{h}.  (2.4)

These vector fields are obtained from the Y_{i} by the following approximation procedure. For any λ > 0, we let Y_{i}^{λ} = λ^{−w_{i}}δ_{λ}Y_{i}, where w_{i} = len(Y_{i}). By Proposition 2.1, we have for any compact set K ⊂ ℝ^{n}

lim_{λ→∞} sup_{x∈K} |Y_{i}^{λ}(x) − Y_{i}^{∞}(x)| = 0,  lim_{λ→∞} sup_{x∈K} |∂_{h}Y_{i}^{λ}(x) − ∂_{h}Y_{i}^{∞}(x)| = 0,  (2.5)

for any h = 1, ..., n, where ∂_{h} acts on the coefficients of the vector fields.

The vector fields X^{λ}, λ > 0, and X^{∞} can be defined starting from any smooth vector field X in ℝ^{n}. First, we define the order of X. We say that Y_{i} has order −w_{i} = −len(Y_{i}) ∈ ℤ and we write ord(Y_{i}) = −w_{i}. If f ∈ C^{∞}(ℝ^{n}) is a smooth function such that the δ-homogeneous polynomial with minimal degree in the Taylor development of f at x = 0 has homogeneous degree equal to k ∈ ℕ ∪ {0}, we say that f has order k and we write ord(f) = k. Clearly, we have ord(fg) = ord(f) + ord(g). Then we let ord(fY_{i}) = ord(f) + ord(Y_{i}) ∈ ℤ. If X = ∑_{i=1}^{n} f_{i}Y_{i}, we let ord(X) = min{ord(f_{i}Y_{i}) : i = 1, ..., n} ∈ ℤ. From ord(Xf) = ord(X) + ord(f), it also follows

ord([X, Y]) = ord(X) + ord(Y),  (2.6)

for smooth vector fields X and Y.

For a smooth vector field X in ℝ^{n} with order ord(X) = k, we let

X^{λ} = λ^{k}δ_{λ}X, X^{∞} = lim_{λ→+∞} X^{λ}.

The limit exists and the convergence is locally uniform. Analogously, we have

lim_{λ→+∞} sup_{x∈K} |∂_{h}X^{λ}(x) − ∂_{h}X^{∞}(x)| = 0,  (2.7)

for any h = 1, ..., n and K ⊂ ℝ^{n} compact set.

The mapping X ↦ X^{∞} preserves the bracket, and precisely we have for any couple of smooth vector fields X and Y

[X, Y]^{∞} = [X^{∞}, Y^{∞}].  (2.8)

END-POINT EQUATIONS 7

This follows from (2.3), (2.6) and (2.7).

By Proposition 2.1 and (2.8), the distribution D^{∞} = span{Y_{1}^{∞}, ..., Y_{m}^{∞}} is still bracket generating.

Lemma 2.2. Assume that condition (1.2) holds and let X, Y ∈ {Y_{1}, ..., Y_{n}}. If X ∈ D_{i}, Y ∈ D_{j} with i, j ≥ 2 and i + j > 4, then [X^{∞}, Y^{∞}] = 0.

Proof. As we have len(X) = i and len(Y) = j, the vector fields X^{∞} and Y^{∞} satisfy and δ_{λ}X^{∞} = λ^{i}X^{∞} and δ_{λ}Y^{∞} = λ^{j}Y^{∞}. From (2.3), it follows that

δ_{λ}[X^{∞}, Y^{∞}] = [δ_{λ}X^{∞}, δ_{λ}Y^{∞}] = λ^{i+j}[X^{∞}, Y^{∞}].  (2.9)

By (1.2), it is [X, Y] ∈ L_{i+j−1} and then k = −ord([X, Y]) ≤ i + j − 1. From

[X, Y]^{∞} = lim_{λ→+∞} λ^{−k}δ_{λ}[X, Y],

we deduce that δ_{λ}[X, Y]^{∞} = λ^{k}[X, Y]^{∞}. Now, from (2.8) and (2.9) it follows that

λ^{k}[X^{∞}, Y^{∞}] = λ^{k}[X, Y]^{∞} = δ_{λ}[X, Y]^{∞} = δ_{λ}[X^{∞}, Y^{∞}] = λ^{i+j}[X^{∞}, Y^{∞}],

for all λ > 0. Then it must be [X^{∞}, Y^{∞}] = 0, because k ≠ i + j.

Let us fix a metric (i.e. a positive quadratic form) g on D. On the distribution D^{λ} = span{Y_{1}^{λ}, ..., Y_{m}^{λ}}, λ > 0, we introduce the metric g^{λ} defined by

g^{λ}(x, Y) = g(δ_{1/λ}(x), λδ_{1/λ}Y),

where x ∈ ℝ^{n} and Y is a vector field in ℝ^{n}. Then, on D^{∞} = span{Y_{1}^{∞}, ..., Y_{m}^{∞}} we introduce the metric g^{∞} defined by

g^{∞}(x, Y^{∞}) = lim_{λ→+∞} g^{λ}(x, Y^{λ}).

Let γ : [−1, 1] → ℝ^{n} be a D-horizontal Lipschitz curve, i.e.

γ̇ = ∑_{i=1}^{m} h_{i}Y_{i}(γ), h_{i} ∈ L^{∞}(−1, 1).

Assume that γ(0) = 0 and that the right and left derivatives of γ at t = 0 exist

γ̇^{+} = lim_{t→0^{+}} \frac{1}{t}γ(t),  γ̇^{−} = lim_{t→0^{−}} \frac{1}{t}γ(t).  (2.10)

We blow up the curve γ at the point 0. For λ > 0 and t ∈ [−λ, λ], let γ^{λ}(t) = δ_{λ}γ(t/λ). The curve γ^{λ} is D^{λ}-horizontal and

γ̇^{λ} = ∑_{i=1}^{m} h_{i}^{λ}Y_{i}^{λ}(γ^{λ}), h_{i}^{λ}(t) := λ^{w_{i}−1}h_{i}(t/λ).

Then, the limit curve γ^{∞} : (−∞, +∞) → ℝ^{n}

γ^{∞}(t) = \{ \begin{matrix}lim_{λ→+∞} γ^{λ}(t) = γ̇^{+}t, & t ≥ 0 \\ lim_{λ→+∞} γ^{λ}(t) = γ̇^{−}t, & t < 0, \end{matrix}  (2.11)

is D^{∞}-horizontal. Here, γ̇^{+}, γ̇^{−} ∈ ℝ^{m} × {0} are fixed vectors in ℝ^{n}.

The length of the curves γ, γ^{λ} : I → ℝ^{n}, λ ∈ ℝ^{+} ∪ {∞}, defined on some interval I = [a, b] ⊂ ℝ is, respectively,

L(γ) = ∫_{I} g(γ(t), γ̇(t))^{1/2}dt, L^{λ}(γ^{λ}) = ∫_{I} g^{λ}(γ^{λ}(t), γ̇^{λ}(t))^{1/2}dt.

We show that, if γ is length minimizing in (ℝ^{n}, D, g), then γ^{∞} is length minimizing in (ℝ^{n}, D^{∞}, g^{∞}). First, we need the following approximation result.

Proposition 2.3. Let κ^{∞} : [0, T] → ℝ^{n} be a D^{∞}-horizontal curve with κ^{∞}(0) = x_{0}. For small enough T > 0, there exist D^{λ}-horizontal curves κ^{λ} : [0, T] → ℝ^{n}, λ ∈ (λ_{0}, +∞) with large enough λ_{0}, such that

i) κ^{λ} → κ^{∞} uniformly as λ → ∞; ii) L^{λ}(κ^{λ}) → L^{∞}(κ^{∞}), as λ → ∞.

Proof. Denote by B = C([0, T]; ℝ^{n}) the Banach space of continuous curves endowed with the uniform norm ∥κ∥ = sup_{t∈[0,T]} |κ(t)|. Starting from κ^{∞}, we define a map F_{λ} : B → B. For measurable functions h_{i} ∈ L^{∞}(0, T), i = 1, ..., m, we have

κ^{∞}(t) = x_{0} + ∫_{0}^{t} ∑_{i=1}^{m} h_{i}Y_{i}^{∞}(κ^{∞})ds,  t ∈ [0, T].  (2.12)

Then we let

F_{λ}(κ) = x_{0} + ∫_{0}^{t} ∑_{i=1}^{m} h_{i}Y_{i}^{λ}(κ)ds, λ ∈ (0, ∞],

where h_{i}’s are the functions in (2.12). By the Contraction Principle, F_{λ} has a fixed point κ^{λ} ∈ B, as soon as we choose a small enough T > 0. By compactness, the family (κ^{λ})_{λ>0} converges to κ^{∞} in B. The proof of ii) is straightforward.

Proposition 2.4. Let γ : [−1, 1] → ℝ^{n} be a length minimizing curve in (ℝ^{n}, D, g) such that the limits in (2.10) exist, and let γ^{∞} : [−1, 1] → ℝ^{n} be the curve (2.11). Then γ^{∞} is length minimizing in (ℝ^{n}, D^{∞}, g^{∞}).

Proof. The length in (ℝ^{n}, D^{λ}, g^{λ}) of the curve γ^{λ}(t) = δ_{λ}γ(t/λ), with t ∈ [−1, 1] and λ > 1, satisfies the identity

∫_{−1}^{1} g^{λ}(γ^{λ}, γ̇^{λ})^{1/2}dt = λ ∫_{−1/λ}^{1/λ} g(γ(t), γ̇(t))^{1/2}dt.

Thus, γ is length minimizing if and only if γ^{λ} is length minimizing in (ℝ^{n}, D^{λ}, g^{λ}) for all λ > 1.

Assume by contradiction that γ^{∞} is not length minimizing. The metric space (ℝ^{n}, d^{∞}), with d^{∞} distance induced by the metric g^{∞}, is locally compact and complete, and then length minimizing curves exist between pair of arbitrary points. Possibly replacing the interval [−1, 1] with an interval [−T, T] with small enough T > 0, we can assume that: 1) there exists a length minimizing curve κ^{∞} : [−1, 1] → ℝ^{n} with κ^{∞}(−1) = γ^{∞}(−1) and κ^{∞}(1) = γ^{∞}(1); 2) there are horizontal curves κ^{λ} : [−1, 1] →

END-POINT EQUATIONS 9

ℝ^{n} in (ℝ^{n}, D^{λ}, g^{λ}) converging uniformly as λ → +∞ to κ^{∞} and with length converging to the length of κ^{∞} (Proposition 2.3). This contradicts, for large enough λ, the length minimality of γ^{λ}. Note that the distance in (ℝ^{n}, D^{λ}, g^{λ}) between κ^{λ}(±1) and γ^{λ}(±1) tends to zero.

We want to prove that γ̇^{+} = γ̇^{−} whenever γ is length minimizing and γ̇^{+}, γ̇^{−} are as in (2.10) with γ̇^{+} ≠ 0 and γ̇^{−} ≠ 0. By contradiction, we suppose this is not the case. Then, either γ̇^{+} = −γ̇^{−}, e.g. γ has a cusp at t = 0, or γ̇^{+} and γ̇^{−} are linearly independent, that is, γ has a corner at t = 0. We can easily exclude the cusp by observing that the corresponding limit curve γ^{∞} in (2.11) would parameterize twice the same interval. We are now left with the case of a corner. In the selection procedure explained just at the beginning of the section, we can choose the vector fields Y_{1} and Y_{2} such that Y_{1}^{∞}(0) = γ̇^{+} and Y_{2}^{∞}(0) = −γ̇^{−} (with the standard identification between vector fields and points of ℝ^{n}). Then we have γ^{∞}(t) = exp(tY_{1}^{∞})(0) for t ≥ 0 and γ^{∞}(t) = exp(−tY_{2}^{∞})(0) for t < 0.

We introduce in ℝ^{n} exponential coordinates of the second type associated with the basis Y_{1}^{∞}, ..., Y_{n}^{∞}. Precisely, define the diffeomorphism Φ : ℝ^{n} → ℝ^{n}

Φ(x) = exp(x_{1}Y_{1}^{∞}) … exp(x_{n}Y_{n}^{∞})(0), x = (x_{1}, ..., x_{n}) ∈ ℝ^{n}.

Denote by Ψ : ℝ^{n} → ℝ^{n} the inverse of Φ, and define the new basis of vector fields

X_{i} = dΨ(Y_{i}^{∞}), i = 1, ..., n.  (2.13)

Remark 2.5. In the new coordinates, the curve γ^{∞} still satisfies γ^{∞}(t) = exp(tX_{1})(0) for t ≥ 0 and γ^{∞}(t) = exp(−tX_{2})(0) for t < 0. Let M_{1} ⊂ ℝ^{n} be the orbit of X_{1} and X_{2}. Since the vector fields X_{1} and X_{2} have polynomial coefficients, it is M_{1} = ℝ^{d} ⊂ ℝ^{n} for some 2 ≤ d ≤ n. Let E be the distribution obtained restricting span{X_{1}, X_{2}} to M_{1}, with the metric h obtained restricting g^{∞} to E. If γ^{∞} is length minimizing in (ℝ^{n}, D^{∞}, g^{∞}), then γ^{∞} is also length minimizing in (ℝ^{d}, E, h). Then we can without loss of generality assume that m = 2 and d = n.

Proposition 2.6. Let D be a rank 2 distribution satisfying (1.1) and (1.2). Then the vector fields X_{1} and X_{2} in (2.13) are of the form

X_{1} = ∂_{1},  X_{2} = ∂_{2} + ∑_{i=3}^{n} P_{i}(x_{1}, x_{2})∂_{i},  (2.14)

where P_{i} are homogeneous polynomials depending only on the variables x_{1} and x_{2} and such that P_{i}(0, x_{2}) = 0, i = 3, ..., n.

Proof. Since we are in exponential coordinates of the second type, we have the identity

x = exp(x_{1}X_{1}) … exp(x_{n}X_{n})(0), x = (x_{1}, ..., x_{n}) ∈ ℝ^{n},  (2.15)

and then we find

X_{1}(x) = \frac{d}{dt} exp(tX_{1})(x)|_{t=0}

= \frac{d}{dt} exp((x_{1} + t)X_{1}) … exp(x_{n}X_{n})(0)|_{t=0} = ∂_{1}.

Let W = W (t, x_{1}, x_{2}) be the vector field defined through the relation

exp(W) = exp(−(x_{2} + t)X_{2}) exp(−x_{1}X_{1}) exp(tX_{2}) exp(x_{1}X_{1}) exp(x_{2}X_{2}).

Here, t, x_{1}, x_{2} are parameters of the vector field W = W (t, x_{1}, x_{2}). By the Campbell-Baker-Hausdorff formula, W is a sum of Lie brackets of the vector fields X_{1} and X_{2} (with coefficients) of length at least 2. By Mitchell-Margulis-Mostows’s theorem (see [Mi], [B] and [MM]), we have a Carnot group structure in ℝ^{n} for which X_{1}, ..., X_{n} are left invariant. Then, Lie brackets of X_{1}, X_{2} with length at least 2 are linear combinations with real coefficients of X_{3}, ..., X_{n}. It follows that

W (t, x_{1}, x_{2}) = ∑_{i=3}^{n} p_{i}(t, x_{1}, x_{2})X_{i},

where the p_{i}’s are homogeneous polynomials in t, x_{1}, x_{2} which satisfy p_{j}(t, 0, x_{2}) = 0 for all i = 3, ..., n. Now, let

Γ(t, x) = exp(tX_{2}) exp(x_{1}X_{1}) … exp(x_{n}X_{n})(0)
= exp(x_{1}X_{1}) exp((x_{2} + t)X_{2}) exp(W) exp(x_{3}X_{3}) … exp(x_{n}X_{n})(0).

By Lemma 2.2, the vector fields W, X_{3}, ..., X_{n} commute, and thus by (2.15)

Γ(t, x) = exp(x_{1}X_{1}) exp((x_{2} + t)X_{2}) exp(W + x_{3}X_{3} + ⋯ + x_{n}X_{n})(0)

= exp(x_{1}X_{1}) exp((x_{2} + t)X_{2}) ∏_{i=3}^{n} exp ((x_{i} + p_{i}(t, x_{1}, x_{2}))X_{i})(0)

= (x_{1}, x_{2} + t, x_{3} + p_{3}(t, x_{1}, x_{2}), …, x_{n} + p_{n}(t, x_{1}, x_{2})).

The polynomials

P_{i}(x_{1}, x_{2}) = \frac{d}{dt}p_{i}(t, x_{1}, x_{2})|_{t=0}

are homogeneous in x_{1}, x_{2} and satisfy P_{i}(0, x_{2}) = 0, i = 3, ..., n. Moreover,

X_{2}(x) = \frac{d}{dt}Γ(x, t)|_{t=0} = ∂_{2} + ∑_{i=3}^{n} P_{i}(x_{1}, x_{2})∂_{i}.

3. End point equations and singularities of corner type

In this section, we prove that length minimizing curves do not have corners. Precisely, we have the following theorem.

Theorem 3.1. Let (M, D, g) be a sub-Riemannian manifold satisfying (1.1) and (1.2). Then a curve with a corner is not length minimizing in (M, D, g).

END-POINT EQUATIONS 11

By Proposition 2.4, it is enough to prove Theorem 3.1 in the case M = ℝ^{n} for the curve γ : [−1, 1] → ℝ^{n} in (1.3). By Remark 2.5, we can without loss of generality assume that m = 2 and D = span{X_{1}, X_{2}}, where, by Proposition 2.6, the vector fields X_{1} and X_{2} are of the form (2.14), with P_{i} = P_{i}(x_{1}, x_{2}) homogeneous polynomials such that P_{i}(0, x_{2}) = 0. Finally, we can without loss of generality fix the metric g on D making X_{1} and X_{2} orthonormal. From now on, our argument does not depend on condition (1.1).

Let γ : [0, 1] → ℝ^{n} be a D-horizontal curve and consider the plane curve κ : [0, 1] → ℝ^{2}, κ = (γ_{1}, γ_{2}). We call κ the horizontal projection of γ. Then we have γ̇ = κ̇_{1}X_{1}(γ)+κ̇_{2}X_{2}(γ) and thus γ̇_{i} = κ̇_{2}P_{i}(κ), i = 3, ..., n. The curve γ is determined by its horizontal projection κ

\begin{aligned}γ_{i}(t) & = γ_{i}(0) + ∫_{κ|_{[0,t]}} P_{i}(x_{1}, x_{2})dx_{2} \\ & = γ_{i}(0) + ∫_{0}^{t} κ̇_{2}(s)P_{i}(κ(s)) ds, i = 3, ..., n. \end{aligned}  (3.1)

Given a Lipschitz curve κ in the plane, a horizontal curve γ defined by (γ_{1}, γ_{2}) = κ and by (3.1) is said to be a lift of κ. If the starting point γ(0) is fixed, we call γ the lift of κ. The length of γ in (ℝ^{n}, D, g) is

L(γ) = ∫_{0}^{1} |κ̇(t)| dt.  (3.2)

In this Section, we prove that the curve γ in (1.3) is not length minimizing in (ℝ^{n}, D, g). We cut the corner by means of triangles

T^{η} = {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1}, x_{2} > 0, x_{1} + x_{2} < η},  (3.3)

where 0 < η < 1/2 is a positive parameter. The boundary ∂T^{η} of T^{η} is oriented according to the rule to start from (0, η) in direction (η, 0). The new curve in the plane has to be lifted in order to have a horizontal curve and the end-point is consequently changed. We correct the error of the end point by means of the following rectangles

\begin{aligned}R_{λ}^{b,ε} & = \{(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1} ∈ (b, b + |ε|^{λ}), x_{2} ∈ (0, ε)\}, \\ S_{λ}^{b,ε} & = \{(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} ∈ (b, b + |ε|^{λ}), x_{1} ∈ (0, ε)\}, \\ P_{λ}^{b,ε} & = \{(x_{1}, x_{2}) ∈ ℝ^{2} : x_{1} ∈ (b, b + ε|ε|^{λ}), x_{2} ∈ (0, |ε|)\}, \\ Q_{λ}^{b,ε} & = \{(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} ∈ (b, b + ε|ε|^{λ}), x_{1} ∈ (0, |ε|)\}. \end{aligned}  (3.4)

Here, 0 < λ < 1, b > 1/2 and ε ∈ ℝ are parameters. We call b position parameter, ε size parameter, and λ induction parameter. Position parameters are needed to solve linear systems of equations (see the proof of Proposition 3.5). Size parameters will eventually depend on η. Induction parameters λ are needed for the convergence of the iterative construction and are fixed in a crucial step of the argument (see Lemma 3.7).

12 GIAN PAOLO LEONARDI AND ROBERTO MONTI

The boundary of the rectangles is oriented according to the following rules. In ∂R_{λ}^{b,ε}, we start from (b, 0) in direction (b, ε); in ∂S_{λ}^{b,ε}, we start from (0, b + |ε|^{λ}) in direction (ε, b + |ε|^{λ}); in P_{λ}^{b,ε}, we start from (b, 0) in direction (b + ε|ε|^{λ}, 0); in Q_{λ}^{b,ε}, we start from (0, b) in direction (0, b + ε|ε|^{λ}). The sign of ε carries information on the orientation.

Starting from the curve γ in (1.3), we can use rectangles R_{λ}^{b,ε} to construct a new horizontal curve: in the plane, we follow γ up to the point (b, 0), then we follow ∂R_{λ}^{b,ε} up to the point (b + |ε|^{λ}, 0), and finally we follow γ again; this plane curve is then lifted to a horizontal curve. Analogously, we can use rectangles P_{λ}^{b,ε} to construct a new horizontal curve in the following way: we follow γ up to the point (b, 0), then we follow the whole boundary ∂P_{λ}^{b,ε} getting again to (b, 0), and finally we follow γ again; this plane curve is then lifted to a horizontal curve. The construction with ∂S_{λ}^{b,ε} (resp. ∂Q_{λ}^{b,ε}) is analogous. Notice that in the case of P_{λ}^{b,ε} and Q_{λ}^{b,ε} the curve γ is deviated along the whole boundary of the rectangle.

Remark 3.2. Deviating γ along a rectangle has a cost of length, which is the difference between the length of the new horizontal curve and the length of γ. This cost can be computed by (3.2). For R_{λ}^{b,ε} and S_{λ}^{b,ε} the cost is 2|ε|, while for P_{λ}^{b,ε} and Q_{λ}^{b,ε} the cost is 2|ε|(1 + |ε|^{λ}) ≤ 4|ε| as soon as |ε| ≤ 1.

Each polynomial P_{i}, i = 3, ..., n, appearing in X_{2} in (2.14) is of the form

P_{i}(x_{1}, x_{2}) = ∑_{α+β=d_{i}} c_{αβ}x_{1}^{α+1}x_{2}^{β},

where d_{i} + 1 is the degree of the polynomial P_{i} and c_{αβ} are constants. From now on, we assume without loss of generality that c_{αβ} ∈ {0, 1}. Recall that each P_{i} must contain x_{1}. The effect on the end-point produced by cutting the curve γ in (1.3) by a triangle T^{η} (and then lifting) or by deviating γ along a rectangle (and then lifting) can be computed by (3.1). This is done in the following Propositions 3.3 and 3.4, which are easily proved by Stokes’s theorem.

Proposition 3.3. Let α, β ≥ 0 be integers. Then we have

∫_{∂T^{η}} x_{1}^{α+1}x_{2}^{β}dx_{2} = C_{αβ}η^{α+β+2},  (3.5)

where C_{αβ} is a constant.

Proposition 3.4. Let α, β ≥ 0 be integers. Then we have

∫_{∂R_{λ}^{b,ε}} x_{1}^{α+1}x_{2}^{β}dx_{2} = −\frac{α + 1}{β + 1}b^{α}|ε|^{λ}ε^{β+1} + O(|ε|^{2λ}ε^{β+1}),  (3.6)
∫_{∂S_{λ}^{b,ε}} x_{1}^{α+1}x_{2}^{β}dx_{2} = b^{β}|ε|^{λ}ε^{α+1} + O(|ε|^{2λ}ε^{α+1}),  (3.7)
∫_{∂P_{λ}^{b,ε}} x_{1}^{α+1}x_{2}^{β}dx_{2} = \frac{α + 1}{β + 1}b^{α}ε|ε|^{β+λ+1} + O(|ε|^{β+2λ+3}),  (3.8)
∫_{∂Q_{λ}^{b,ε}} x_{1}^{α+1}x_{2}^{β}dx_{2} = −b^{β}ε|ε|^{α+λ+1} + O(|ε|^{α+2λ+3}).  (3.9)

Here, O(ε)/ε is bounded as ε → 0.

The curve γ can be deviated along two or more disjoint rectangles (3.4). The total effect on the end-point is the sum of the effects computed in Proposition 3.4. We refer to this property as to the additivity of perturbations.

Consider the set P of all monomials x_{1}^{α+1}x_{2}^{β} appearing in the coefficients P_{i}(x_{1}, x_{2}) of X_{2} in (2.14). The same monomial could appear in different coefficients. Let

N = max {min{α, β} : x_{1}^{α+1}x_{2}^{β} ∈ P} .

We partition P into subsets P_{k}, Q_{k}, with k = 0, 1, ..., N. We let x_{1}^{α+1}x_{2}^{β} ∈ P_{k} if k = min{α, β} and β ≤ α. We let x_{1}^{α+1}x_{2}^{β} ∈ Q_{k} if k = min{α, β} and α < β. Then we have the partition

P = ⋃_{k=0}^{N} P_{k} ∪ Q_{k}.

Let π_{k} = ♯(P_{k}) and ω_{k} = ♯(Q_{k}) be the cardinality of P_{k} and Q_{k}. Without loss of generality, we assume that

P_{k} = {x_{1}^{k+h}x_{2}^{k} : h = 1, ..., π_{k}}, Q_{k} = {x_{1}^{k+1}x_{2}^{k+h} : h = 1, ..., ω_{k}}.

On the i-th coordinate, i = 3, ..., n, the cut T^{η} produces the total error

T_{i}^{η} = ∫_{∂T^{η}} P_{i}(x_{1}, x_{2}) dx_{2}.  (3.10)

Instead of correcting the error T_{i}^{η} for each coordinate i, we correct the errors associated with the monomials in P_{k} and Q_{k} for each k = 0, 1, ..., N. More precisely, let us consider the following vectors of errors

T_{P_{k}}^{η} = ∫_{∂T^{η}} x_{1}^{k}x_{2}^{k}(x_{1}, ..., x_{1}^{π_{k}}) dx_{2}, T_{Q_{k}}^{η} = ∫_{∂T^{η}} x_{1}^{k+1}x_{2}^{k}(x_{2}, ..., x_{2}^{ω_{k}}) dx_{2}.  (3.11)

For any k = 0, 1, ..., N, we correct the errors T_{P_{k}}^{η} (and the errors T_{Q_{k}}^{η}) by solving one system of equations. On iterating this procedure, we correct the whole error (3.10) for all i = 3, ..., n.

In general, we call a π_{k}-dimensional vector of real numbers (η_{1}, ..., η_{π_{k}}) the error of P_{k} and denote by u_{k} = max{|η_{h}| : h = 1, ..., π_{k}} the size of the error. Analogously, we

14 GIAN PAOLO LEONARDI AND ROBERTO MONTI

call a ω_{k}-dimensional vector of real numbers (η_{1}, ..., η_{ω_{k}}) the error of Q_{k} and denote by v_{k} = max{|η_{h}| : h = 1, ..., ω_{k}} the size of the error. In our argument, it is sufficient to estimate the size of errors. Then we introduce the space

E = {w = (u_{0}, v_{0}, ..., u_{N}, v_{N}) ∈ ℝ^{2(N+1)} : u_{k}, v_{k} ≥ 0},

and we endow E with the partial ordering ≤ defined by w ≤ w′ if and only if u_{k} ≤ u′_{k} and v_{k} ≤ v′_{k} for all k = 0, 1, ..., N.

A correction is a map I : E → E describing the propagation of errors produced by the correction of the error of some P_{k}’s and Q_{k}’s using rectangles (3.4) with suitable parameters. If the initial error is w ∈ E, we denote by Λ(I, w) the cost of length of correction I, which is the cost of length of all the rectangles employed in the correction.

If x_{1}^{α+1}x_{2}^{β} ∈ P_{0}, then β = 0 and the most efficient way (i.e. with the smallest size parameter |ε|) to correct an error of P_{0} is to use rectangles R_{λ}^{b,ε}. Note that rectangles P_{λ}^{b,ε} are forbidden. Indeed, by (3.8) with β = 0, we see that, in order to correct by such rectangles an error of size η^{2} (by (3.5)), the length of the curve increases at least of an amount η^{\frac{2}{2+λ}}, which is too large (even with λ = 0). Analogously, the most efficient way to correct an error of Q_{0} is to use rectangles S_{λ}^{b,ε}. Note that, also in this case, rectangles Q_{λ}^{b,ε} are forbidden because the cost of length would be too large.

Proposition 3.5. Let λ > 0. There exist C > 0 and a correction I_{0} : E → E of the error of P_{0} such that for all w ∈ E with small enough |w| we have

I_{0}(w) ≤ (0, v_{0} + Cu0^{\frac{λ+2}{λ+1}} , ..., u_{k} + Cu0^{\frac{k+λ+1}{λ+1}} , v_{k} + Cu0^{\frac{k+λ+2}{λ+1}} , ...) ,  (3.12)

and with a cost of length satisfying

Λ(I_{0}, w) ≤ Cu0^{\frac{1}{1+λ}} .  (3.13)

Analogously, there is a correction J_{0} : E → E of the error of Q_{0} such that

J_{0}(w) ≤ (u_{0} + Cv_{0}, 0, ..., u_{k} + Cv0^{\frac{k+λ+1}{λ+1}} , v_{k} + Cv0^{\frac{k+λ+1}{λ+1}} , ...) ,  (3.14)

and with a cost of length satisfying

Λ(J_{0}, w) ≤ Cv0^{\frac{1}{1+λ}} .  (3.15)

Proof. Let I = {1, ..., m} where m = π_{0}. Without loss of generality, we can assume that monomials in P_{0} are of the form x_{1}^{α_{i}+1} with α_{i} = i − 1, i ∈ I (we have β = 0). We denote by η_{I} = (η_{1}, ..., η_{m}) the error of P_{0} that must be corrected, with size u_{0} = max_{i∈I} |η_{i}|. Consider disjoint rectangles R_{i} = R_{λ}^{b_{i},ε_{i}} with i ∈ I, where the position parameters b_{i} are required to satisfy the conditions

1/2 < b_{i} < 1,  b_{i+1} − b_{i} ≥ c_{0},  (3.16)

for some small universal constant c_{0} > 0. This choice is possible for small size parameters ε_{i}. Denote by ε_{I} the ordered vector of the ε_{i}’s and denote by Δ_{j}(ε_{I}) the effect produced according to (3.6) by these rectangles on x_{1}^{α_{j}+1} ∈ P_{0}. By the additivity property of perturbations and by (3.6) with β = 0, we have

Δ_{j}(ε_{I}) = ∑_{i∈I} ( − (α_{j} + 1)b_{i}^{α_{j}}ε_{i}|ε_{i}|^{λ} + O(ε_{i}|ε_{i}|^{2λ})).

Finally, let Δ(ε_{I}) be the ordered vector of the Δ_{j}(ε_{I}), j ∈ I. In order to correct the error η_{I}, the parameters ε_{I} must solve the system of non linear equations −Δ(ε_{I}) = η_{I}. This system can be linearized at the origin with respect to the new variables ε′_{i} = ε_{i}|ε_{i}|^{λ}. The matrix Γ of the linearized system is

Γ = ( \begin{matrix}1 & 1 & ... & 1 \\ 2b_{1} & 2b_{2} & ... & 2b_{π_{0}} \\ ⋮ & ⋮ & & ⋮ \\ mb_{1}^{m−1} & mb_{2}^{m−1} & ... & mb_{m_{0}}^{m−1} \end{matrix} ) .

By (3.16), the Vandermonde determinant satisfies

det(Γ) = m! ∏_{1≤i<j≤m}(b_{j} − b_{i}) ≥ C_{0} > 0  (3.17)

for some universal constant C_{0}. Thus, the non linear system −Δ(ε_{I}) = η_{I} has one solution ε_{I} for any datum η_{I} in a neighborhood of the origin, and by (3.17) the solution satisfies

|ε_{I}| ≤ C_{1}|η_{I}|^{\frac{1}{1+λ}}  (3.18)

for some constant C_{1} > 0.

Now we have a correction I_{0} : E → E of P_{0}. We prove that it satisfies (3.12). The rectangles R_{i} with parameters ε_{I} produce an error on monomials which are not in P_{0}. Denote by Δ_{P_{k}}^{P_{0}} with k ≥ 1 (resp. Δ_{Q_{k}}^{P_{0}} with k ≥ 0) the size of the error produced on P_{k} (resp. Q_{k}). By formula (3.6) with α = β = k and by (3.18), we find

Δ_{P_{k}}^{P_{0}} ≤ C_{2}|ε_{I}|^{k+λ+1} ≤ C_{3}|η_{I}|^{\frac{k+λ+1}{λ+1}}  k ≥ 1.  (3.19)

By formula (3.6) with α = k and β = k + 1, we find

Δ_{Q_{k}}^{P_{0}} ≤ C_{4}|ε_{I}|^{k+λ+2} ≤ C_{5}|η_{I}|^{\frac{k+λ+2}{λ+1}} ,  k ≥ 0.  (3.20)

From (3.19) and (3.20), we obtain estimate (3.12) for correction I_{0}.

The cost of length of one rectangle R_{λ}^{b,ε} is 2|ε| (see Remark 3.2). Then, by (3.18), estimate (3.13) immediately follows.

Now we correct the error of Q_{0}. Let J = {1, ..., ω_{0}}. Without loss of generality, we can assume that monomials in Q_{0} are of the form x_{1}x_{2}^{β_{i}} where β_{i} = i with i ∈ J (we have α = 0). We use rectangles S_{i} = S_{λ}^{b_{i},ε_{i}} with i ∈ J. Let ε_{J} be the ordered vector of the ε_{i} with i ∈ J and let Δ_{j}(ε_{J}) be the perturbation produced by the rectangles S_{i} on the monomial x_{1}x_{2}^{β_{j}} ∈ Q_{0}. From (3.7) with α = 0, we have

Δ_{j}(ε_{J}) = ∑_{i∈J} (b_{i}^{β_{j}}ε_{i}|ε_{i}|^{λ} + O(ε_{i}|ε_{i}|^{2λ})) .

16 GIAN PAOLO LEONARDI AND ROBERTO MONTI

Denote by Δ(ε_{J}) the ordered vector of the Δ_{j}(ε_{J}), with j ∈ J. Let η_{J} = (η_{1}, ..., η_{ω_{0}}) be the error of Q_{0} with size v_{0} = max_{j∈J} |η_{j}|. A linearization argument similar to the one in the first part of the proof shows that the system of non linear equations −Δ(ε_{J}) = η_{J} has one solution ε_{J} for all η_{J} in a neighborhood of the origin. Moreover, this solution satisfies

|ε_{J}| ≤ C_{6}|η_{J}|^{\frac{1}{1+λ}} .  (3.21)

We have a correction J_{0} : E → E of Q_{0}. We show that it satisfies (3.14). Denote by Δ_{P_{k}}^{Q_{0}} with k ≥ 0 (resp. Δ_{Q_{k}}^{Q_{0}} with k ≥ 1) the size of the error produced by the rectangles S_{i} on P_{k} (resp. Q_{k}). By formula (3.7) with α = β = k (resp. with α = k and β = k + 1) we have the estimates

Δ_{P_{k}}^{Q_{0}} ≤ C_{7}|ε_{J}|^{k+λ+1} ≤ C_{8}|η_{J}|^{\frac{k+λ+1}{λ+1}} k ≥ 0,  (3.22)
Δ_{Q_{k}}^{Q_{0}} ≤ C_{9}|ε_{J}|^{k+λ+1} ≤ C_{10}|η_{J}|^{\frac{k+λ+1}{λ+1}} ,  k ≥ 1.  (3.23)

From (3.22) and (3.23), the estimate (3.14) immediately follows.

The cost of length of one rectangle S_{λ}(b, ε) is 2|ε|. Then, estimate (3.15) follows from (3.21).

In order to correct the error P_{h} with h ≥ 1 we use rectangles P_{λ}^{b,ε}. Note that, e.g. in the relevant case h = 1, rectangles R_{λ}^{b,ε} are forbidden. The reason is that the sign of each component of the error can be both positive and negative, whereas in formula (3.6) with β = h = 1 the right hand side is non positive for any small ε. In order to correct the error of Q_{h} with h ≥ 1, we employ rectangles Q_{λ}^{b,ε}. Rectangles S_{λ}^{b,ε} are, in this case, forbidden.

Proposition 3.6. Let λ > 0. There exist C_{0} and a correction I_{h} : E → E of the error of P_{h}, h ≥ 1, such that for all w ∈ E with small enough |w| we have

I_{h}(w) ≤ (..., 0, v_{h} + Cuh^{\frac{λ+h+3}{λ+h+2}} , ..., u_{k} + Cuh^{\frac{λ+k+2}{λ+h+2}} , v_{k} + Cuh^{\frac{λ+k+3}{λ+h+2}} , ...) .  (3.24)

In the round brackets, 0 is at position u_{h} and it is understood k ≠ h. The cost of length of correction I_{h} satisfies

Λ(I_{h}, w) ≤ Cuh^{\frac{1}{λ+h+2}} .  (3.25)

Analogously, there is a correction J_{h} : E → E of the error of Q_{h}, h ≥ 1, such that

J_{h}(w) ≤ (..., u_{h} + Cv_{h}, 0, ..., u_{k} + Cvh^{\frac{λ+k+2}{λ+h+2}} , v_{k} + Cvh^{\frac{λ+k+2}{λ+h+2}} , ...) .  (3.26)

Here, 0 is at position v_{h} and k ≠ h. The cost of length of correction J_{h} satisfies

Λ(J_{h}, w) ≤ Cvh^{\frac{1}{λ+h+2}} .  (3.27)

END-POINT EQUATIONS 17

Proof. Letting I_{h} = {1, ..., π_{h}}, we can assume that the monomials in P_{h} are of the form x_{1}^{α_{i}+1}x_{2}^{β_{i}} where β_{i} = h and α_{i} = h + i − 1, with i ∈ I_{h}. Consider rectangles P_{λ}^{b_{i},ε_{i}} with i ∈ I_{h}. Denote by ε_{I_{h}} the ordered vector of the parameters ε_{i} with i ∈ I_{h} and denote by Δ_{j}(ε_{I_{h}}) the perturbation of the rectangles P_{λ}^{b_{i},ε_{i}} on the monomial x_{1}^{α_{j}+1}x_{2}^{h}. From (3.8) with β = h we have

Δ_{j}(ε_{I_{h}}) = ∑_{i∈I_{h}} (−\frac{α_{j} + 1}{h + 1}b_{i}^{α_{j}}ε_{i}|ε_{i}|^{λ+h+1} + O(|ε_{i}|^{2λ+h+3})) .

Denote by Δ(ε_{I_{h}}) the ordered vector of the Δ_{j}(ε_{I_{h}}) with j ∈ I_{h}. Let η_{I_{h}} = (η_{1}, ..., η_{π_{h}}) be the error of P_{h} with size u_{h} = max_{i∈I_{h}} |η_{i}|. In order to correct this error, the parameters ε_{I_{h}} must solve the non linear system −Δ(ε_{I_{h}}) = η_{I_{h}}. A linearization argument similar to the one in the proof of Proposition 3.5 shows the the solution satisfies

|ε_{I_{h}}| ≤ C_{1}|η_{I_{h}}|^{\frac{1}{λ+h+2}} .  (3.28)

This correction I_{h} of the error of P_{h} produces an additional error of size Δ_{P_{k}}^{P_{h}} on P_{k} with k ≠ h, and an additional error of size Δ_{Q_{k}}^{P_{h}} on Q_{k} with k ≥ 0. On choosing in formula (3.8) the values α = β = k for P_{k} and α = k and β = k + 1 for Q_{k} we find, respectively,

Δ_{P_{k}}^{P_{h}} ≤ C_{2}|ε_{I_{h}}|^{λ+k+2} ≤ C_{3}|η_{I_{h}}|^{\frac{λ+k+2}{λ+h+2}} , k ≠ h  (3.29)
Δ_{Q_{k}}^{P_{h}} ≤ C_{4}|ε_{I_{h}}|^{λ+k+3} ≤ C_{5}|η_{I_{h}}|^{\frac{λ+k+3}{λ+h+2}} , k ≥ 0.  (3.30)

From (3.29) and (3.30), we obtain estimate (3.24) for I_{h}.

By Remark 3.2, the cost of length of each rectangle P_{λ}^{b_{i},ε_{i}} is bounded by 4|ε_{I_{h}}|. Then estimate (3.25) follows from (3.28).

Letting J_{h} = {1, ..., ω_{h}}, we can assume that the monomials in Q_{h} are of the form x_{1}^{α_{i}+1}x_{2}^{β_{i}} where α_{i} = h and β_{i} = h + i, with i ∈ I_{h}. Let η_{J_{h}} = (η_{1}, ..., η_{ω_{h}}) be the error of Q_{h} with size v_{h} = max_{i∈J_{h}} |η_{i}|. We employ rectangles Q_{λ}^{b_{i},ε_{i}} with i ∈ J_{h}. With the usual notation, from (3.9) we have

Δ_{j}(ε_{J_{h}}) = ∑_{i∈J_{h}} (−b_{i}^{β_{j}}ε_{i}|ε_{i}|^{λ+h+1} + O(|ε_{i}|^{2λ+h+3})) .

The solution of the non linear system −Δ(ε_{J_{h}}) = η_{J_{h}} satisfies

|ε_{J_{h}}| ≤ C_{6}|η_{J_{h}}|^{\frac{1}{λ+h+2}} .  (3.31)

Moreover, from (3.9) with α = β = k and with α = k, β = k+1, we find, respectively,

Δ_{P_{k}}^{Q_{h}} ≤ C_{7}|ε_{J_{h}}|^{λ+k+2} ≤ C_{8}|η_{J_{h}}|^{\frac{λ+k+2}{λ+h+2}} ,  (3.32)
Δ_{Q_{k}}^{Q_{h}} ≤ C_{9}|ε_{J_{h}}|^{λ+k+2} ≤ C_{10}|η_{J_{h}}|^{\frac{λ+k+2}{λ+h+2}} .  (3.33)

Thus we get estimate (3.26) for the correction J_{h}. Estimates (3.27) follows from (3.31).

18 GIAN PAOLO LEONARDI AND ROBERTO MONTI

Now we define the parameters needed in the iteration argument. Let 0 < λ̄ < 1 be a number to be fixed later and consider N + 1 real numbers

0 < λ_{0} < λ_{1} < … < λ_{N} < λ̄.  (3.34)

Define by induction the parameters

σ_{0} = 1, σ_{h+1} = \frac{λ_{h} + h + 3}{λ_{h} + h + 2}σ_{h},  0 ≤ h ≤ N − 1.  (3.35)

The vector (σ_{0}, σ_{1}, …, σ_{N}) is an array of increasing exponents which controls the decay of errors in the correction procedure of Theorem 3.11. In the next lemma, we list the properties concerning σ_{h} that are needed in our argument.

Lemma 3.7. i) Given 0 ≤ k < h ≤ N, we have

\frac{σ_{h}}{σ_{k}} ≥ \frac{λ + h + 2}{λ + k + 2}, for all λ ≥ λ_{h−1}.  (3.36)

Given 0 ≤ h < k ≤ N, we have

\frac{σ_{k}}{σ_{h}} ≤ \frac{λ + k + 2}{λ + h + 2}, for all 0 < λ ≤ λ_{h}.  (3.37)

Both in (3.36) and in (3.37), equality holds if and only if λ = λ_{k} and h = k + 1. ii) There exists ϱ > 1 such that

σ_{h} \frac{λ_{h} + j + 2}{λ_{h} + h + 2} ≥ ϱσ_{j}, for all 0 ≤ j < h ≤ N.  (3.38)

iii) For all 1 < τ < 4/3, there exists 0 < λ̄ < 1 in (3.34) such that

\frac{λ_{0} + k + 1}{λ_{0} + 1} ≥ τ \frac{λ_{0} + k + 2}{λ_{0} + 2}, for all k ≥ 1,  (3.39)

and

k + 1 ≥ τσ_{k}, for all k ≥ 1,  (3.40)

Proof. First, note that for any h ≥ 0, the function

λ ↦ \frac{λ + h + 3}{λ + h + 2} is strictly decreasing for λ > 0.  (3.41)

We prove (3.36) by induction on h = k + 1, ..., N. For h = k + 1, (3.36) follows from (3.35) and (3.41), with equality if and only if λ = λ_{k}. We assume that (3.36) holds for h and we prove it for h + 1. By (3.35) and λ ≥ λ_{h} > λ_{h−1}, we have

\frac{σ_{h+1}}{σ_{k}} = \frac{σ_{h}(λ_{h} + h + 3)}{σ_{k}(λ_{h} + h + 2)} ≥ \frac{(λ + h + 2)(λ_{h} + h + 3)}{(λ + k + 2)(λ_{h} + h + 2)} > \frac{λ + h + 3}{λ + k + 2}.

The last inequality follows from (3.41). The proof of (3.37) is similar and is omitted.

Inequality (3.38) follows from (3.36) and from the discussion of equality case. Inequality (3.39) follows from

min_{1≤k≤N} \frac{λ_{0} + k + 1}{λ_{0} + k + 2} = \frac{λ_{0} + 2}{λ_{0} + 3} ≥ τ \frac{λ_{0} + 1}{λ_{0} + 2}.

The latter inequality holds as soon as 0 < λ̄ < 1 is small enough. Similarly, we get also (3.40).

The constant ϱ > 1 in (3.38) exists thanks to strict inequalities in (3.34). It is needed in order to avoid loops of errors in the iterative argument of Theorem 3.11. The constant 1 < τ < 4/3 provides the estimate for the cost of length in Theorem 3.12. From now on, the parameters σ_{h} satisfy all statements in Lemma 3.7. The parameter 0 < λ̄ < 1 in (3.34) is fixed at the end of the argument. We need this last observation.

Remark 3.8. If x = x(λ) and y = y(λ) are continuous functions of λ = (λ_{0}, ..., λ_{N}), we write x ∼ y for x(0) = y(0). With this notation, we have

σ_{h} ∼ \frac{h + 2}{2},  h ≥ 0.  (3.42)

This can be proved by induction using (3.35).

We correct simultaneously the error of P_{0} and of Q_{0}, by iterating countably many times the corrections I_{0} and J_{0} of Proposition 3.5. This is the inductive base of our argument. In the proof we use the following

Lemma 3.9. Let φ : [0, δ_{0}] → [0, δ_{0}] be a function such that φ(δ) ≤ δ/2 for all δ ∈ [0, δ_{0}]. Then, for any ξ > 0 we have

∑_{ν=1}^{+∞} φ^{ν}(δ)^{ξ} ≤ δ^{ξ},  δ ∈ [0, δ_{0}],  (3.43)

as soon as δ_{0} > 0 is small enough. Here, φ^{ν} denotes the iteration φ ∘ ... ∘ φ, ν times.

Proposition 3.10. Let 1 < τ < 4/3. There exist δ_{0} > 0, C > 0 and a correction L_{0} : E → E of the error of P_{0} and Q_{0} satisfying

L_{0}(w) ≤ (0, 0, ..., u_{k} + Cu_{0}^{τσ_{k}}, v_{k} + Cu_{0}^{τσ_{k}}, ...)  (3.44)

for all w ∈ E such that

v_{0} ≤ u_{0} ≤ δ_{0}.  (3.45)

Moreover, the cost of length of correction L_{0} satisfies

Λ(L_{0}, w) ≤ Cu0^{\frac{1}{1+λ_{0}}} .  (3.46)

Proof. We use Proposition 3.5 with λ = λ_{0}. Without loss of generality, we assume

that the constant C in Proposition 3.5 satisfies C ≤ 1. Letting z_{0} = v_{0} + u0^{\frac{λ_{0}+2}{λ_{0}+1}} , the composition of corrections I_{0} and J_{0} satisfies

J_{0} ∘ I_{0}(w) ≤ (z_{0}, 0, ..., u_{k} + u0^{\frac{λ_{0}+k+1}{λ_{0}+1}} + z0^{\frac{λ_{0}+k+1}{λ_{0}+1}} , v_{k} + u0^{\frac{λ_{0}+k+2}{λ_{0}+1}} + z0^{\frac{λ_{0}+k+1}{λ_{0}+1}} , ...) .

By (3.45), we have z_{0} ≤ 2u_{0}. Then, the iteration K_{0} = (J_{0} ∘ I_{0})^{2} satisfies

K_{0}(w) ≤ (4u0^{\frac{λ_{0}+2}{λ_{0}+1}} , 0, ..., u_{k} + C_{1}u0^{\frac{λ_{0}+k+1}{λ_{0}+1}} , v_{k} + C_{1}u0^{\frac{λ_{0}+k+1}{λ_{0}+1}} , ...) ,

for some constant C_{1} > 1. Note that, by (3.37) with h = 0 and λ = λ_{0} and by (3.39), we have

σ_{k} ≤ \frac{λ_{0} + k + 2}{λ_{0} + 2} ≤ \frac{λ_{0} + k + 1}{τ(λ_{0} + 1)}, k ≥ 1.

Letting φ(δ) = 4δ^{\frac{λ_{0}+2}{λ_{0}+1}} , the iterations K_{0}^{ν}, with ν ∈ ℕ, satisfy

K_{0}^{ν}(w) ≤ (φ^{ν}(u_{0}), 0, ...,u_{k} + C_{1} ∑_{μ=0}^{ν−1} φ^{μ}(u_{0})^{τσ_{k}}, v_{k} + C_{1} ∑_{μ=0}^{ν−1} φ^{μ}(u_{0})^{τσ_{k}}, ...),

where φ^{0}(u_{0}) stands for u_{0}. Then, by (3.43), for small enough δ_{0} > 0 we can define the correction L_{0} : E → E

L^{0}(w) = lim_{ν→+∞} K_{0}^{ν}(w) ≤ (0, 0, ..., u^{k} + C^{2}u_{0}^{τσ_{k}}, v^{k} + C^{2}u_{0}^{τσ_{k}}, ...),  (3.47)

for some C_{2} > 0.

This proves the existence of a correction L_{0} satisfying (3.44). Some explanations are in order. The correction J_{0} ∘ I_{0} consists in placing rectangles R_{λ_{0}}^{b,ε} on the x_{1} axis to correct the error of P_{0} and then in placing rectangles S_{λ_{0}}^{b,ε} on the x_{2} axis to correct the error of Q_{0}. After repeating this construction twice, a new error of P_{0} with size 4u_{0}^{(λ_{0}+2)/(λ_{0}+1)} appears. We iterate the process countably many times. During the process, the parameter λ_{0} > 0 is fixed, but size parameters ε are (more than) geometrically decreasing and position parameters b are chosen in such a way that all the countably many rectangles used are pairwise disjoint and confined in a bounded region. In particular, position parameters b must satisfy at each stage condition (3.16). This is possible, if λ_{0} > 0 and size parameters are small enough.

We check (3.46). By (3.18) and (3.21), the cost of length of correction J_{0} ∘ I_{0} is estimated by

Λ(J_{0} ∘ I_{0}, w) ≤ C_{1}u0^{\frac{1}{1+λ_{0}}} .

The same estimate holds for K_{0}. Then, we have by (3.43)

Λ(L_{0}, w) ≤ C_{4}(u0^{\frac{1}{1+λ_{0}}} + ∑_{ν=1}^{+∞} φ^{ν}(u_{0})^{\frac{1}{1+λ_{0}}} ) ≤ C_{5}u0^{\frac{1}{1+λ_{0}}} .  (3.48)

This finishes the proof.

The iteration argument of Proposition 3.5 is used freely in the next theorem, which is the core of the paper.

END-POINT EQUATIONS 21

Theorem 3.11. For any h = 0, 1, ..., N, there exist a constant δ_{h} > 0, C > 0 and a correction L_{h} : E → E of the errors of P_{0}, Q_{0}, ..., P_{h}, Q_{h} satisfying

L_{h}(w) ≤ ( \underbrace{0, ..., 0}_{2h+2}, ..., u_{k} + Cu_{0}^{σ_{k}}, v_{k} + Cu_{0}^{σ_{k}}, ...),  (3.49)

for all w ∈ E such that u_{0} ≤ δ_{h} and

max{u_{k}, v_{k}} ≤ u_{0}^{σ_{k}}, 0 ≤ k ≤ h.  (3.50)

The cost of length of correction L_{h} satisfies

Λ(L_{h}, w) ≤ Cu0^{\frac{σ_{h}}{λ_{h}+h+2}} .  (3.51)

Proof. Our argument is by induction on h = 0, 1, ..., N. The case h = 0 is proved in Proposition 3.10 (with even better estimates). We assume that we have a correction L_{h−1} satisfying (3.49) and (3.51) under condition (3.50) with h − 1 replacing h, and we prove the existence of the correction L_{h}.

We preliminary estimate the iteration K_{h} = (J_{h} ∘ I_{h})^{2}. Without loss of generality, we assume that the constant C in Proposition 3.6 satisfies C ≤ 1. By (3.24) and (3.26) with λ = λ_{h}, we have

J_{h} ∘ I_{h}(w) = (..., z_{h}, 0, ..., u_{k} + uh^{\frac{λ_{h}+k+2}{λ_{h}+h+2}} + zh^{\frac{λ_{h}+k+2}{λ_{h}+h+2}} , v_{k} + uh^{\frac{λ_{h}+k+3}{λ_{h}+h+2}} + zh^{\frac{λ_{h}+k+2}{λ_{h}+h+2}} , ...),

where we let z_{h} = v_{h} + uh^{\frac{λ_{h}+h+3}{λ_{h}+h+2}} . Here, 0 is at position v_{h} and k ≠ h. By (3.50), we have z_{h} ≤ 2u_{0}^{σ_{h}} and then we find

J_{h} ∘ I_{h}(w) ≤ (..., 2u_{0}^{σ_{h}}, 0, ..., u_{k} + C_{1}u_{0}^{σ_{h} \frac{λ_{h}+k+2}{λ_{h}+h+2}} , v_{k} + C_{1}u_{0}^{σ_{h} \frac{λ_{h}+k+2}{λ_{h}+h+2}} , ...),

for some C_{1} > 0. The iteration, then, satisfies

K_{h}(w) ≤ (..., 4u_{0}^{σ_{h} \frac{λ_{h}+h+3}{λ_{h}+h+2}} , 0, ..., u_{k} + C_{2}u_{0}^{σ_{h} \frac{λ_{h}+k+2}{λ_{h}+h+2}} , v_{k} + C_{2}u_{0}^{σ_{h} \frac{λ_{h}+k+2}{λ_{h}+h+2}} , ...), (3.52)

for some new constant C_{2} > 0. Note that, here, we have σ_{h} \frac{λ_{h}+h+3}{λ_{h}+h+2} = σ_{h+1}, by (3.35). We estimate the cost of length Λ(K_{h}, w). From (3.25) with λ = λ_{h} and (3.50)

Λ(I_{h}, w) ≤ uh^{\frac{1}{λ_{h}+h+2}} ≤ u0^{\frac{σ_{h}}{λ_{h}+h+2}} .

Analogously, from (3.27) with λ = λ_{h} e from z_{h} ≤ 2u_{0}^{σ_{h}} we find

Λ(J_{h}, I_{h}(w)) ≤ zh^{\frac{1}{λ_{h}+h+2}} ≤ 2u0^{\frac{σ_{h}}{λ_{h}+h+2}} .

It follows that for some constant C_{3} > 0, we have

Λ(K_{h}, w) ≤ C_{3}u0^{\frac{σ_{h}}{λ_{h}+h+2}} .  (3.53)

Let us consider the composition K_{h} ∘ L_{h−1}. By (3.52), (3.49) for h − 1, and (3.50) we have

K_{h} ∘ L_{h−1}(w) ≤ (..., 4u_{0}^{σ_{h+1}}, 0, ..., ū_{k} + C_{2}u_{0}^{σ_{h} \frac{λ_{h}+k+2}{λ_{h}+h+2}} , v̄_{k} + C_{2}u_{0}^{σ_{h} \frac{λ_{h}+k+2}{λ_{h}+h+2}} , ...), (3.54)

22 GIAN PAOLO LEONARDI AND ROBERTO MONTI

where

ū_{k} = u_{k} + u_{0}^{σ_{k}}, v̄_{k} = v_{k} + u_{0}^{σ_{k}}, for h < k ≤ N,
ū_{k} = 0,  v̄_{k} = 0,  for 0 ≤ k < h.

We estimate the exponents in (3.54). By (3.38) and (3.37), we have

\begin{matrix}σ_{h} \frac{λ_{h} + j + 2}{λ_{h} + h + 2} ≥ ϱσ_{j} & for 0 ≤ j < h, \\ σ_{h} \frac{λ_{h} + k + 2}{λ_{h} + h + 2} ≥ σ_{k} & for h < k ≤ N, \end{matrix}

where ϱ > 1 is a constant. Then for some C_{4} > 0, we get from (3.54)

K_{h} ∘ L_{h−1}(w) ≤ (..., C_{2}u_{0}^{ϱσ_{j}}, C_{2}u_{0}^{ϱσ_{j}}, ..., 4u_{0}^{σ_{h+1}}, 0, ..., u_{k} + C_{4}u_{0}^{σ_{k}}, v_{k} + C_{4}u_{0}^{σ_{k}}, ...), (3.55)

where 0 ≤ j < h and h < k ≤ N.

Let us consider the iteration (K_{h} ∘ L_{h−1})^{ν}, ν ∈ ℕ. Letting φ(δ) = C_{2}δ^{ϱ}, we have that the iterated sequence φ^{ν}(δ), ν ∈ ℕ, is infinitesimal as soon as 0 ≤ δ < δ_{h} with small enough δ_{h}, because ϱ > 1. Moreover, by Lemma 3.9 we have

∑_{ν=1}^{+∞} φ^{ν}(u_{0})^{σ_{k}} ≤ u_{0}^{σ_{k}}, h < k ≤ N.

Also iterations of ψ(δ) = 4δ^{σ_{h+1}/σ_{h}} are infinitesimal. Then we have a correction L_{h} : E → E of the errors of P_{0}, Q_{0}, ..., P_{h}, Q_{h} satisfying

L_{h}(w) ≤ lim_{ν→+∞}(K_{h} ∘ L_{h−1})^{ν}(w) ≤ ( \underbrace{0, ..., 0}_{2h+2}, ..., u_{k} + C_{5}u_{0}^{σ_{k}}, v_{k} + C_{5}u_{0}^{σ_{k}}, ...), (3.56)

for some constant C_{5} > 0. This proves (3.49).

We prove that correction L_{h} satisfies (3.51). From (3.53), (3.51) with h − 1 and (3.50) with k = h, we have

Λ(K_{h} ∘ L_{h−1}, w) ≤ Cu0^{\frac{σ_{h−1}}{λ_{h−1}+h+1}} + C_{3}u0^{\frac{σ_{h}}{λ_{h}+h+2}} ≤ C_{6}u0^{\frac{σ_{h}}{λ_{h}+h+2}} ,

because, by (3.34) and (3.35), we have

\frac{σ_{h−1}}{λ_{h−1} + h + 1} ≥ \frac{σ_{h}}{λ_{h} + h + 2}.

Then, by Lemma 3.9 we have for some C_{7} > 0

Λ(L_{h}, w) ≤ C_{7}u0^{\frac{σ_{h}}{λ_{h}+h+2}} .

This is (3.51).

We conclude our main argument. Let (ℝ^{n}, D, g) be the sub-Riemannian manifold introduced at the beginning of this section. By Section 2, Theorem 3.1 is a corollary of the following

END-POINT EQUATIONS 23

Theorem 3.12. The curve γ : [−1, 1] → ℝ^{n} in (1.3) is not length minimizing in (ℝ^{n}, D, g).

Proof. Let κ = (γ_{1}, γ_{2}) be the horizontal projection of γ onto the x_{1}x_{2}-plane and let κ^{η} : [−1, 1] → ℝ^{2} be the plane curve obtained cutting κ along the triangle T^{η} in (3.3), with 0 < η < 1/2. Let γ^{η} be the horizontal lift of κ^{η} starting from γ(−1), according to (3.1). By (3.2), we have

L(γ) − L(γ^{η}) = (2 − \sqrt{2})η.  (3.57)

By (3.10), the error at the end-point at each coordinate i = 3, ..., n is

γ_{i}^{η}(1) = T_{i}^{η} = ∫_{∂T^{η}} P_{i}(x_{1}, x_{2})dx_{2}.

Let u_{h}^{η} and v_{h}^{η} be the size of the errors T_{P_{h}}^{η} and T_{Q_{h}}^{η} in (3.11). This size is given by the first component of the integrals in (3.11), that are computed in (3.5). Up to a multiplicative constant (which can be assumed to be less than 1), we have u_{h}^{η} = η^{2h+2} and v_{h}^{η} = η^{2h+3}. Let w^{η} = (u_{0}^{η}, v_{0}^{η}, ..., u_{N}^{η}, v_{N}^{η}).

We first correct the error η^{2} of P_{0} and the error η^{3} of Q_{0}. We use correction L_{0} : E → E of Proposition 3.10. After this correction, we have by (3.44) the new error

L_{0}(w^{η}) ≤ (0, 0, ..., η^{2k+2} + η^{2τσ_{k}}, η^{2k+3} + η^{2τσ_{k}}, ...) ∈ E.

Here, it is 1 ≤ k ≤ N and 1 < τ < 4/3. By (3.46), the cost of length of L_{0} satisfies

Λ(L_{0}, w^{η}) ≤ C_{1}η^{\frac{2}{1+λ_{0}}} .  (3.58)

Let w̄^{η} = (..., η^{2τσ_{k}}, η^{2τσ_{k}}, ...), with k = 0, 1, ..., N. By (3.40), we have L_{0}(w^{η}) ≤ 2w̄^{η} and, moreover, w̄^{η} satisfies (3.50). By Theorem 3.11, we can correct the error L_{0}(w^{η}) by the correction L_{N} : E → E. By (3.51), the cost of length is at most

Λ(L_{N}, 2w̄^{η}) ≤ C_{2}η^{\frac{2τσ_{N}}{λ_{N} +N+2}} ,  (3.59)

where C_{2} > 0 is a constant which does not depend on η.

Thus, there exists a horizontal curve γ̄^{η} : [−1, 1] → ℝ^{n} such that γ̄^{η}(−1) = γ(−1), γ̄^{η}(1) = γ(1) and such that, by (3.58) and (3.59),

L(γ̄^{η}) − L(γ^{η}) ≤ C_{3} (η^{\frac{2}{1+λ_{0}}} + η^{\frac{2τσ_{N}}{λ_{N} +N+2}} ) ≤ C_{4}η^{\frac{2τσ_{N}}{λ_{N} +N+2}} .  (3.60)

With the notation of Remark 3.8, we have, by (3.42),

\frac{2τσ_{N}}{λ_{N} + N + 2} ∼ τ with 1 < τ < \frac{4}{3}.  (3.61)

Thus, by continuity we get

\frac{2τσ_{N}}{λ_{N} + N + 2} > 1,

as soon as 0 < λ̄ < 1 in (3.34) is small enough. At this point λ̄ is fixed once for all. Now, from (3.57) and (3.60), we obtain L(γ̄^{η}) < L(γ) for small η > 0 and the Theorem is proved.

24 GIAN PAOLO LEONARDI AND ROBERTO MONTI

4. Examples of distributions with smooth geodesics

We discuss some examples of sub-Riemannian manifold with a rank 2 distribution. We study singular curves and we compute the geodesics equations for length minimizing curves which are not singular. A general theory for rank 2 distributions is developed by Liu and Sussmann in [LS] (see in particular Sections 6 and 7). Here, we give a self-contained presentation of the topic taking the Lagrangian view point.

We consider a rank 2 distribution D in ℝ^{n} spanned by vector fields X_{1}, X_{2} as in (2.14). We restrict our analysis to vector fields with coefficients P_{j} = P_{j}(x_{1}, x_{2}), j = 3, ..., n, that are C^{1} functions depending only on x_{1}, x_{2}. On D we fix the metric g making X_{1} and X_{2} orthonormal. Let γ : [0, 1] → ℝ^{n} be a D-horizontal curve with horizontal projection κ : [0, 1] → ℝ^{2}. We assume that κ is parameterized by constant speed (i.e. |κ̇| = L a.e. for some L ≠ 0). Let ϑ_{ij} ∈ C_{0}^{∞}(0, 1) with i = 1, 2 and j = 1, ..., k = n − 2, be test functions and consider the matrix

Θ = (\begin{matrix}ϑ_{11} & ⋯ & ϑ_{1k} \\ ϑ_{21} & ⋯ & ϑ_{2k} \end{matrix} ) = (\begin{matrix}Θ_{1} \\ Θ_{2} \end{matrix}) .

Here, Θ_{i} is the row (ϑ_{i1}, ..., ϑ_{ik}), i = 1, 2. For τ ∈ ℝ^{k}, let κ^{τ} : [0, 1] → ℝ^{2} be the plane curve κ^{τ} = κ + Θτ, and denote by γ^{τ} the horizontal lift of κ^{τ} with γ^{τ}(0) = γ(0) (τ, κ and κ^{τ} are columns and Θτ is a matrix-column product). By (3.1), we have for i = 3, ..., n

γ_{i}^{τ}(t) = γ_{i}(0) + ∫_{0}^{t} P_{i}(κ^{τ})κ̇_{2}^{τ} ds, t ∈ [0, 1].

Note that γ^{τ} also depends on Θ. For fixed Θ and γ, define the end-point map Φ(·; Θ; γ) : ℝ^{k} → ℝ^{k}

Φ(τ; Θ; γ) = (γ_{3}^{τ}(1), ..., γ_{n}^{τ}(1)) ,

i.e. Φ_{i−2}(τ; Θ; γ) = γ_{i}^{τ}(1), for i = 3, ..., n. We take derivatives in τ of Φ and we denote by ∇_{τ}Φ_{i−2} the gradient of Φ_{i−2}. Integrating by parts, we find

\begin{aligned}∇_{τ}Φ_{i−2}(τ; Θ; γ) & = ∫_{0}^{1} \{∇P_{i}(κ^{τ})Θ κ̇_{2}^{τ} + P_{i}(κ^{τ})Θ̇_{2}\} dt \\ & = ∫_{0}^{1} \{∇P_{i}(κ^{τ})Θ κ̇_{2}^{τ} − ∇P_{i}(κ^{τ})κ̇^{τ} Θ_{2}\}dt. \end{aligned}

In the integral, ∇P_{i} is the gradient of P_{i} with respect to the variables x_{1}, x_{2}, and ∇P_{i}Θ is a row-matrix product. At τ = 0 we have

\begin{aligned}∇_{τ}Φ_{i−2}(0; Θ; γ) & = ∫_{0}^{1} \{∇P_{i}(κ)Θκ̇_{2} − ∇P_{i}(κ)κ̇ Θ_{2}\}dt \\ & = ∫_{0}^{1} ∂_{1}P_{i}(κ)\{κ̇_{2}Θ_{1} − κ̇_{1}Θ_{2}\}dt. \end{aligned}  (4.1)

Finally, the Jacobian matrix of Φ is

J_{τ}Φ(τ; Θ; γ) = ( ∇_{τ}Φ_{1}(τ; Θ; γ) ; ⋮ ; ∇_{τ}Φ_{n−2}(τ; Θ; γ) )

Definition 4.1. We say that a D-horizontal curve γ : [0, 1] → ℝ^{n} is singular if

det (J_{τ}Φ(0; Θ; γ)) = 0  (4.2)

for all Θ = (ϑ_{ij}) with ϑ_{ij} ∈ C^{∞}(0, 1), i = 1, 2 and j = 1, ..., k.

Let K : ℝ^{k} × ℝ^{2} → ℝ be the function

K(μ, x) = ∑_{i=1}^{k} μ_{i}∂_{1}P_{i+2}(x),

where μ = (μ_{1}, ..., μ_{k}) ∈ ℝ^{k} and x ∈ ℝ^{2}. The connection between K and singular curves is explained in the following

Proposition 4.2. A D-horizontal curve γ : [0, 1] → ℝ^{n} with horizontal projection κ is singular if and only if there are constants μ ∈ ℝ^{k} with μ ≠ 0 such that

K(μ, κ(t)) = 0,  t ∈ [0, 1].  (4.3)

Proof. If (4.3) is verified for some μ ≠ 0, then the vectors in (4.1) are linearly dependent and then condition (4.2) holds.

Now assume that (4.2) holds for some Lipschitz curve κ : [0, 1] → ℝ^{2} with |κ̇| = L a.e. for some constant L ≠ 0. Let E ⊂ [0, 1] be a negligible set such that κ̇(t) exists, |κ̇(t)| = L and t is a Lebesgue point of κ̇ for all t ∈ [0, 1] \ E.

Given t_{1}, ..., t_{k} ∈ [0, 1]\E, there exists a sequence of Θ^{h} = (ϑ_{ij}^{h}) with ϑ_{ij}^{h} ∈ C_{0}^{∞}(0, 1) such that the vector valued functions κ̇_{2}Θ_{1}^{h} − κ̇_{1}Θ_{2}^{h} ∗-weakly converge to a k-vector of Dirac measures supported in t_{1}, ..., t_{k}. Letf_{i}(t) = ∂_{1}P_{i+2}(κ(t)), i = 1, ..., k. From (4.1) and (4.2) with Θ = Θ^{h}, letting h → ∞ we get

Δ(t_{1}, ..., t_{k}) = det ( \begin{matrix}f_{1}(t_{1}) & … & f_{1}(t_{k}) \\ ⋮ & ... & ⋮ \\ f_{k}(t_{1}) & … & f_{k}(t_{k}) \end{matrix} ) = 0  (4.4)

for all t_{1}, ..., t_{k} ∈ [0, 1] \ E. By continuity, (4.4) holds for all points in [0, 1].

In view of (4.4), let 1 ≤ h ≤ k − 1 be the largest integer such that there exist t_{1}, ..., t_{h} ∈ [0, 1] for which

Δ(t_{1}, ..., t_{h}) = det ( \begin{matrix}f_{1}(t_{1}) & … & f_{1}(t_{h}) \\ ⋮ & ... & ⋮ \\ f_{h}(t_{1}) & … & f_{h}(t_{h}) \end{matrix} ) ≠ 0.

If there is no such an index h, then we have in particular f_{1}(t) = 0 for all t ∈ [0, 1], and the claim is proved. Otherwise, it follows that

Δ(t_{1}, ..., t_{h}, t) = det ( \begin{matrix}f_{1}(t_{1}) & … & f_{1}(t_{h}) & f_{1}(t) \\ ⋮ & ... & ⋮ & ⋮ \\ f_{h}(t_{1}) & … & f_{h}(t_{h}) & f_{h}(t) \\ f_{h+1}(t_{1}) & … & f_{h+1}(t_{h}) & f_{h+1}(t) \end{matrix} ) = 0,

for all t ∈ [0, 1]. Developing the determinant with respect to the last column, we find

∑_{i=1}^{h+1} μ_{i}f_{i}(t) = 0,  t ∈ [0, 1],  (4.5)

for constants μ_{1}, ..., μ_{h+1} such that μ_{h+1} = Δ(t_{1}, ..., t_{h}) ≠ 0.

Proposition 4.3. Let γ : [0, 1] → ℝ^{n} be a horizontal curve which is length minimizing in (ℝ^{n}, D, g). If γ is not singular, then γ ∈ C^{∞}(0, 1) and its horizontal projection κ solves the the system of equations

κ̈ = K(μ, κ)κ̇^{⊥},  (4.6)

for constants μ ∈ ℝ^{k}. Here, κ^{⊥} = (−κ_{2}, κ_{1}).

Proof. Let Θ be a matrix of test functions such that det (J_{τ}Φ(0; Θ; γ)) ≠ 0. For ε ∈ ℝ and test functions ξ_{1}, ξ_{2} ∈ C_{0}^{∞}(0, 1) define the curve κ^{τ,ε} : [0, 1] → ℝ^{2}

κ^{τ,ε} = κ + Θτ + εΞ, Ξ := (\begin{matrix}ξ_{1} \\ ξ_{2} \end{matrix}) .

Let γ^{τ,ε} : [0, 1] → ℝ^{n} be the lift of κ^{τ,ε} such that γ^{τ,ε}(0) = γ(0).

Let Ψ(·; Ξ; γ) : ℝ^{k+1} → ℝ^{k}, k = n − 2, be the map

Ψ(τ, ε; Ξ; γ) = (γ_{3}^{τ,ε}(1), ..., γ_{n}^{τ,ε}(1)) .

The Jacobian matrix J := J_{τ}Ψ(0; Ξ; γ) = J_{τ}Φ(0; Θ; γ) is invertible. By the implicit function theorem, for any Ξ there exists a C^{1} function ε ↦ τ(ε) ∈ ℝ^{k} defined in some interval I = (−δ, δ) such that Ψ(τ(ε), ε; Ξ; γ) = Ψ(0; Ξ; γ) = (γ_{3}(1), ..., γ_{n}(1)) for all ε ∈ I. Moreover, at ε = 0 we have

τ′(0) = −J^{−1} \frac{∂Ψ(0; Ξ; γ)}{∂ε}.  (4.7)

Here, with abuse of notation, ∂Ψ/∂ε is a k-column vector.

All the curves γ^{τ(ε),ε} start from γ(0) and arrive at γ(1). Letting

L(ε; Ξ) = ∫_{0}^{1} |κ̇ + Θ̇τ(ε) + εΞ̇| dt,

the length minimality of γ implies that L′(0; Ξ) = 0, for all test functions Ξ. Using |κ̇| = L and (4.7), we find (here ⟨·, ·⟩ is the standard scalar product in ℝ^{m}, m ≥ 2)

L′(0; Ξ) = \frac{1}{L} ∫_{0}^{1} \{⟨κ̇, Ξ̇⟩ + ⟨κ̇, Θ̇τ′(0)⟩\} dt

= \frac{1}{L} ∫_{0}^{1} \{⟨κ̇, Ξ̇⟩ − ⟨κ̇, Θ̇J^{−1} \frac{∂Ψ(0; Ξ; γ)}{∂ε}⟩\} dt

= \frac{1}{L} ∫_{0}^{1} \{⟨κ̇, Ξ̇⟩ − ⟨Aκ̇, \frac{∂Ψ(0; Ξ; γ)}{∂ε}⟩\} dt,

where A = (Θ̇J^{−1})^{T}. A computation analogous to (4.1) yields

\frac{∂Ψ_{i−2}(0; Ξ; γ)}{∂ε} = ∫_{0}^{1} ∂_{1}P_{i}(κ)\{κ̇_{2}ξ_{1} − κ̇_{1}ξ_{2}\}dt, i = 3, ..., n,  (4.8)

and, letting μ = ∫_{0}^{1} Aκ̇ dt, we get the equation

∫_{0}^{1} {⟨κ̇, Ξ̇⟩ + K(μ, κ)⟨κ̇^{⊥}, Ξ⟩} dt = 0

for all test functions Ξ. By a standard argument, it follows that κ ∈ C^{∞}(0, 1) and integrating by parts we find (4.6).

Example 4.4. If the functions P_{j} = P_{j}(x_{1}, x_{2}) are polynomials, then all length minimizing curves in (ℝ^{n}, D, g) are of class C^{1}.

Indeed, let γ be a length minimizing curve. If γ is not singular, then it is of class C^{∞} by Proposition 4.3. If γ is singular, by Proposition 4.2 its horizontal projection κ is contained in the algebraic curve

Γ = {x ∈ ℝ^{2} : K(μ, x) = 0}  (4.9)

for some μ ∈ ℝ^{k}, μ ≠ 0. Assume that γ passes through a singular point of the curve. As noted in Section 2, γ does not have a cusp at the singular point. By Theorem 3.12, γ does not have a corner, either. Then, by known properties of algebraic curves, γ is at least of class C^{1} at the singular point.

Example 4.5. If the functions P_{j} = P_{j}(x_{1}, x_{2}) are polynomials such that ∂_{1}P_{j} has degree at most 2 for all j = 3, ..., n, then all length minimizing curves in (ℝ^{n}, D, g) are, in fact, of class C^{∞}. Indeed, the algebraic curve Γ in (4.9) is a conic, and by Theorem 3.12, κ must be contained in a smooth piece of the conic.

Example 4.6. An example covered by Example 4.5 are Carnot groups with rank 2 and step r ≤ 4. Indeed, generators of the Lie algebra for the free nilpotent homogeneous group with rank 2 and step 4 are the vector fields in ℝ^{8} (see e.g. [GG])

X_{1} = ∂_{1}, X_{2} = ∂_{2} + x_{1}∂_{3} + x_{1}^{2}∂_{4} + x_{1}x_{2}∂_{5} + x_{1}^{3}∂_{6} + x_{1}^{2}x_{2}∂_{7} + x_{1}x_{2}^{2}∂_{8}.

It follows that length minimizing curves in Carnot groups with rank 2 and step r ≤ 4 endowed with a left invariant metric are of class C^{∞}.

References

[AG] A. Agrachev, J.-P. Gauthier, On the subanalyticity of Carnot-Caratheodory distances, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), no. 3, 359–382.

[AS] A. Agrachev, Y. Sachkov, Control theory from the geometric viewpoint. Encyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer-Verlag, Berlin, 2004.

[ASa] A. Agrachev, A. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. Henri Poincaré, 13 n. 16 (1996), 635–690.

[B] A. Bellaïche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian geometry, 1–78, Progr. Math., 144, Birkhäuser, Basel, 1996.

[GK] Chr. Golé, R. Karidi, A note on Carnot geodesics in nilpotent Lie groups, J. Dynam. Control Systems 1 (1995), no. 4, 535–549.

[GMT] E. Gonzales, U. Massari, I. Tamanini, Boundaries of prescribed mean curvature, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 4 (1993), no. 3, 197–206.

[GG] M. Grayson, R. Grossman, Models for free nilpotent Lie algebras, J. Algebra 135 (1990), no. 1, 177–191.

[LS] W. Liu, H. Sussmann, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564.

[MM] G. A. Margulis, G. D. Mostow, Some remarks on the definition of tangent cones in a Carnot-Carathéodory space, J. Anal. Math. 80 (2000), 299–317.

[Mi] J. Mitchell, On Carnot-Carathéodory metrics, J. Differ. Geom. 21 (1985), 35–45.

[M1] R. Montgomery, Abnormal minimizers, SIAM J. Control Optim., 32 (1994), 1605–1620.

[M2] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, 91. American Mathematical Society, Providence, RI, 2002.

[S] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom., 24 (1986), 221–263. [Corrections to “Sub-Riemannian geometry”, J. Differential Geom., 30 (1989), 595–596].

Gian Paolo Leonardi: Dipartimento di Matematica Pura ed Applicata, Università di Modena e Reggio Emilia, Via Campi, 213/b, 41100 Modena, Italy

  E-mail address: leonardi.gianpaolo@unimo.it

Roberto Monti: Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Trieste, 63, 35121 Padova, Italy

  E-mail address: monti@math.unipd.it