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On the connectedness of the space of codimension one foliations on a closed 3-manifold

Abstract

We study the topology of the space of smooth codimension one foliations on a closed 3-manifold. We regard this space as the space of integrable plane fields included in the space of all smooth plane fields. It has been known since the late 60’s that every plane field can be deformed continuously to an integrable one, so the above inclusion induces a surjective map between connected components. We prove that this map is actually a bijection.

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Acknowledgments

I am grateful beyond words to Emmanuel Giroux for his invaluable advice and support, through all the stages of this work. This text also owes a lot to all those who gave me the opportunity to present and discuss my work, and more generally who showed their interest and shared their thoughts. I apologize for not being able to thank them all personally here. I am also very thankful to the referees for their many helpful comments and suggestions, especially regarding the Appendix.

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Correspondence to Hélène Eynard-Bontemps.

Appendix: flexibility of almost integrable plane fields

Appendix: flexibility of almost integrable plane fields

This section is devoted to the proof of Proposition 3.9, which states that any two homotopic almost integrable plane fields are actually homotopic through almost integrable plane fields. Actually, in order to prove Theorem B, we need a more general statement, replacing the parameter space [0, 1] and its boundary \(\{0,1\}\) by a compact finite dimensional polyhedron K and a closed subpolyhedron L of K (typically, \(K = {\mathbb {D}}^n\) and \(L = {\mathbb {S}}^{n-1}\)).

We will use the following vocabulary. A K-plane field \(\xi \) on a manifold M is a family \(\xi _t,\,t\in K\), of plane fields on M. Now given a subset \(X \subset K \times M\), we say that a K-plane field \(\xi \) is integrable on X if for every \(t\in K\), the plane field \(\xi _t\) is integrable on \(X_t = X \cap (\{t\} \times M)\). In practice, X is often of the form \((K \times A) \cup (L \times M)\), where A is a subset of M. We say that a K-plane field \(\xi \) is almost horizontal on a collection of balls \(B\subset M\) if, for every \(t\in K\), the plane field \(\xi _t\) is almost horizontal on B. Finally, we say that a K-plane field \(\xi \) is \((K' \times B)\)-almost integrable if for every \(t\in K'\subset K\), the plane field \(\xi _t\) is B-almost integrable.

Recall that given a subset A of a topological space, the notation \(\mathop {\mathrm {Op}}\nolimits (A)\) refers to a small nonspecified open neighborhood of A.

Proposition 6.1

Consider a closed 3-manifold M, a collection of balls B in M, a compact finite dimensional polyhedron K and a closed subpolyhedron L of K. Let \(\xi \) be an \((L \times B)\)-almost integrable K-plane field on M, almost horizontal on B. There exists a K-plane field \( {\bar{\xi }} \) on M with the following properties:

  1. 1.

    \( {\bar{\xi }} \) is homotopic to \(\xi \) relative to \((K \times \mathop {\mathrm {Op}}\nolimits B) \cup (L \times M)\);

  2. 2.

    \( {\bar{\xi }} \) is \((K \times \overline{B})\)-almost integrable for some collection of balls \(\overline{B}\) containing B.

In order to deform plane fields to integrable ones, Thurston initiated the use of triangulations. He demonstrated the effectiveness of his idea in [2224]. Eliashberg then adapted the techniques of [23] in [8] to deform plane fields to contact structures, and extended them to families of plane fields depending on any number of parameters. In return, Proposition 6.1 and its proof are modeled on part of [8], namely Lemma 3.2.1 and its proof, which relies on Sects. 2.3 and 2.4 of the same paper. Our aim here is mainly to detail and complete Eliashberg’s arguments (see in particular Remark 6.5). We also refer the reader to the book [10] by Geiges for further details about the complete argument of [8].

We will now give an outline of the proof of Proposition 6.1, which takes up the entire Appendix. In particular, we will try to emphasize the difference between the nonparametric and multiparametric construction of almost-integrable plane fields (cf. “Proof of Lemma 6.2...”), and to motivate our choice to give a full proof of the multiparametric version, including a tiresome induction argument, rather than restrict to the one-parameter case which would convey most of the ideas.

Reduction to \({\mathbb {R}}^3\). First, in Sect. 6.1, we cover M with finitely many charts to reduce to a problem in \({\mathbb {R}}^3\). The rest of the Appendix is devoted to the analogue of Proposition 6.1 in \({\mathbb {R}}^3\), namely Lemma 6.2, whose proof we now outline.

The nonparametric case. Before dealing with families of plane fields, we first recall Thurston’s strategy to make one plane field \(\xi \) almost integrable. The starting point is to construct a triangulation in “good position” (or “general position” in Thurston’s words) with respect to \(\xi \), meaning basically that the direction of \(\xi \) is “almost constant” on each 3-simplex (this can be ensured simply by taking the triangulation fine enough) and that the faces and edges are transverse to \(\xi \) (this is achieved by “jiggling” the previous triangulation).

Good position makes it “easy” to make \(\xi \) integrable in a neighborhood of the 2-skeleton and to pick this neighborhood so that \(\xi \) is almost horizontal on each ball of the complement. More precisely, one first makes \(\xi \) integrable in a neighborhood of every vertex, then every edge and finally every face. The deformations near all simplices of a given dimension should be thought of as simultaneous, the tricky part being of course to guarantee the compatibility of deformations performed near adjacent simplices. This is made possible by the existence, near every simplex \(\sigma \), of a vector field \(\nu \) tangent to \(\xi \) and transverse to \(\sigma \). The deformation then consists in keeping \(\xi \) unchanged on \(\sigma \) and making it invariant under \(\nu \) in a neighborhood of \(\mathop {\mathrm {Int}}\sigma \), covered by a flow box of \(\nu \) with base \(\mathop {\mathrm {Int}}\sigma \) (cf. Lemma 6.6 for a generalized quantitative version of this process). Since \(\xi \) is already integrable near \(\partial \sigma \) by the previous step, it is already invariant under \(\nu \) there and thus remains unchanged, which guarantees the global coherence of these local perturbations.

Note the importance of the transversality condition on the triangulation. If the triangulation was not in good position with respect to \(\xi \) (but still sufficiently fine), one could still find, for every face \(\sigma \), a vector field \(\nu \) tangent to \(\xi \) with a flow box covering a neighborhood of \(\sigma \), and make \(\xi \) invariant under this flow. But any flow line leaving and reentering the neighborhood of \(\partial \sigma \) would be a potential obstruction to keeping \(\xi \) unchanged in this neighborhood, which was our guarantee for the global coherence of the perturbations. So one would have to deal with these “special faces” first, like “big vertices”, before carrying on with the other (actual) vertices, faces and edges. This is a problem we have to face in our parametric situation.

Proof of Lemma 6.2 : meaning of the Key Lemma 6.4 and of the “curvature” Lemma 6.3. Indeed, what we want, in order to prove Lemma 6.2, is to deform an entire family \(\xi _t,\,t\in K\), of plane fields to make them all integrable outside the same balls. To that end, we must use the same triangulation for every value of the parameter. But we cannot expect a single triangulation to be in good position with respect to every plane field (the triangulation can be fine enough that every \(\xi _t\) is almost constant near each 3-simplex, but since the direction of \(\xi _t\) varies with t, there is no hope to fulfil the transversality condition in general). The common triangulation we use is a rescaling by a small factor d of a specific triangulation \(\Delta \) of \({\mathbb {R}}^3\) defined in Sect. 6.3 (its main features will be presented below). The choice, or rather the existence of a proper scaling factor d plays an important and elaborate part in the proof. We try to clarify what underlies this choice by writing in italics the relevant parts of the following outline.

The Key Lemma 6.4 (Sect. 6.3) claims that, for any d sufficiently small (so that each plane field is “almost constant” near each simplex), our K-plane field can indeed be made integrable in a neighborhood of the 2-skeleton of the rescaled triangulation \(d\Delta \). As mentioned earlier, the fact that the triangulation is not in good position with respect to every plane field makes this already somewhat harder than in the nonparametric case: some \(\xi _t\) might be tangent to some face \(\sigma \) of some 3-simplex \(\tau \) and these parameters and “special” simplices (cf. Definition 6.17) have to be dealt with in a particular way.

But perhaps more importantly, parameters (and in particular situations like the one above) make it harder to guarantee the almost horizontality of all plane fields on the complement of a common neighborhood of the 2-skeleton: without further precautions, a plane field \( {\tilde{\xi }} _t\) obtained after deforming a plane field \(\xi _t\) as above might have infinitely many points of tangency with the boundary sphere of some randomly embedded ball in \(\tau \). This will not happen, however, if the sphere is convex enough compared to the variations of \( {\tilde{\xi }} _t\), as explained in Sect. 6.2 (cf. Lemma 6.3). But remember this sphere must lie in the neighborhood of the 2-skeleton where the K-plane field has been made integrable. This is why the Key Lemma 6.4 has to quantify the size of this neighborhood, namely a \(\mu d\)-neighborhood (cf. 2. in the Key Lemma), for some \(\mu \) depending only on the initial K-plane field, but not on the scaling factor d. That way, once we have the Key Lemma, choosing d small enough once again, we can make the spheres as curved as we like.

At that point, we use a key property of the model triangulation \(\Delta \): its 3-simplices are all copies of a finite number of model simplices. One can embed a strictly convex sphere in the \(\mu \)-neighborhood of the boundary of each of them. The principal curvatures of these model spheres are bounded below by some positive number k. Then the spheres we embed in each 3-simplex of \(d\Delta \) are simply scaled copies of these model spheres and thus have principal curvatures bounded below by k / d, which can be made as big as we like by taking d sufficiently small.

But the deformed K-plane field given by the Key Lemma depends on the triangulation, and thus on the choice of d. So when we shrink d to increase the curvature of the spheres we can embed, we change \( {\tilde{\xi }} _t\), and possibly its \(C^1\) norm, which determines the minimal curvature guaranteeing the almost horizontality.... So, in short, we need to make sure that the variations of the plane fields resulting from the Key Lemma remain bounded regardless of the scaling factor d, hence the need for point 3. in the Key Lemma.

Now that we have explained the content of the Key Lemma and how it combines with Lemma 6.3 to produce plane fields that are indeed almost integrable, and thus prove Lemma 6.2, we can go on with the outline of the proof of the Key Lemma itself, which is carried out in Sects. 6.4 and 6.5.

Proof of the Key Lemma I: the deformation model. Again, the aim is roughly to show that, for any d small enough, one can make a given K-plane field integrable on a neighborhood of the 2-skeleton of \(d\Delta \) whose diameter does not depend on d up to scaling, keeping control (again independent of d) on the \(C^1\) norm of the resulting plane fields. To that end, the idea is, like in Thurston’s process (cf. the nonparametric case), to apply a local deformation model repeatedly (namely near each simplex of the triangulation, “special” or not).

But we want quantitative control on the resulting object, that depends neither on the number of times we applied the model nor on the specific simplices to which we applied it. So we need a quantitative deformation model which, for a sufficiently large (but necessarily restricted) class of plane fields, tells us how to make them integrable and gives uniform control on the size of the corresponding perturbation. This is the content of Lemma 6.6 to which Sect. 6.4 is devoted. No scaling parameter d is involved there, since Lemma 6.6 is precisely intended to provide bounds independent of d in the end. This lemma is really a statement about plane fields defined near a simplex of the “big” triangulation \(\Delta \) and will be applied to the plane fields of the Key Lemma defined near a simplex of \(d\Delta \) only after a rescaling by a factor 1 / d.

Now the difficulty of the proof of the Key Lemma consists in reducing to situations where the deformation model is indeed applicable (which, in particular, requires the triangulation to be fine enough) and to do things in the right order so that each step is compatible with the previous ones.

Proof of the Key Lemma II: triangulation of the parameter space and induction. Again, given a scaling factor d (which will be chosen a posteriori), the idea is to perturb every plane field \(\xi _t,\,t\in K\), in a neighborhood of every simplex of the 2-skeleton of \(d\Delta \) to make it integrable there, and to do this continuously with respect to t. The problem is that, given t, the order in which one must deal with the simplices if one wants to avoid incompatibilities depends on the position of \(\xi _t\) with respect to the triangulation. Indeed, as mentioned above (cf. Proof of 6.2...), one has to start with special faces to which \(\xi _t\) is “almost tangent” (d being assumed small enough here that the direction of each \(\xi _t\) on a given simplex is “almost constant”). But this, of course, depends on t, so the order in which the deformation is conducted also does, which is a bad start if we want a deformation continuous in t...

To deal with this issue, we start by triangulating the parameter space itself finely enough so that if one plane field \(\xi _{t_0}\) is almost tangent to some face of the “spatial” triangulation, then all other \(\xi _t\)’s are, for t in the same simplex of the “temporal” triangulation as \(t_0\). Thus, each simplex of the “temporal” triangulation has its own set of special faces. All of this must of course be explicitly quantified. In particular, the “almost tangency” is defined so that a single plane field cannot be “almost tangent” to two adjacent faces (which would be a problem because we want to be able to deform it near all special faces simultaneously and independently). This uses a second key property of the triangulation \(\Delta \): the existence of a uniform lower bound on the angles between adjacent simplices (not contained in one another), “almost tangency”, and thus “special faces”, being defined in terms of that bound.

One then proceeds by induction on the successive skeleta of K (cf. Lemma 6.19). Our reasons for carrying out the induction explicitly for any K, rather than reducing to the case where K is made of a single simplex (cf. [8, p. 632]) or where K is one-dimensional, are the following. At each step, we need the new special faces (i.e. the ones associated to the plane fields obtained after the previous step), for any given simplex \(K_*\) of K, to still be disjoint. Therefore, we must, at each step, keep control on the angle between the “old” plane fields and the new ones (cf. last condition in Lemma 6.19). This (and simply the possibility to apply the deformation Lemma near each simplex) puts constraints on the choice of d. Furthermore, recall that we want a bound on the \(C^1\) norm of the resulting plane fields independent from the choice of the scaling factor d. But as we will see in more detail in Sect. 6.4, the local deformation Lemma 6.6 only provides a bound on the \(C^k\) norm of the resulting plane field in terms of the \(C^{k+1}\) norm of the initial one. Thus the induction works basically as follows:

  • We start with a K-plane field \(\xi \) (automatically \(C^{n+1}\)-bounded on any compact set, where n denotes the dimension of K).

  • For any d small enough, one can perturb \(\xi _t\) for t in a neighborhood of the 0-skeleton of K (leaving \(\xi _t\) unchanged for t outside a bigger neighborhood) to make it integrable first near special 2-simplices, then inductively over 0-, 1- and normal 2-simplices of \(d\Delta \). This is done by applying the deformation model to \(\xi _t\) in a neighborhood of each simplex (or rather to a rescaled version of it by a factor 1 / d). We then obtain a K-plane field \(\xi ^1\) and a \(C^n\) bound, proportional to d, on the rescaling of \(\xi ^1\) by a factor 1 / d. Provided d was chosen small enough, the deformed plane fields are arbitrarily \(C^0\)-close to the old ones. In particular, this implies that special simplices for the new ones are still disjoint.

  • Shrinking d in the previous step a posteriori if necessary (the above being valid  for any d below some given bound determined by the initial K-plane field), we continue with the 1-skeleton of K, carrying out the previous step relative to the boundary of the 1-skeleton, i.e. to the 0-skeleton of K. We obtain a K-plane field \(\xi ^2\) with a \(C^{n-1}\) bound on the rescaling of \(\xi ^1\) by a factor 1 / d, and again a bound on the angle by which the plane fields have been modified.

  • We continue on \(K^2,\,K^3, \dots , K^n\) and obtain \(\xi ^3,\,\xi ^4, \dots , \xi ^n=: {\bar{\xi }} \) with \(C^{n-2},\,C^{n-3},\dots , C^1\) bounds proportional to d on their rescalings. Since the \(C^1\) norm of \( {\bar{\xi }} \) is 1 / d times the \(C^1\) norm of its rescaling (cf. Subsection 6.5.4), this gives the desired \(C^1\) control on \( {\bar{\xi }} \).

For simplicity, we have not mentioned here the matter of the size of the neighborhood of the 2-skeleton on which the plane fields are made integrable, but this too must be controlled from one step to the next.

Throughout the appendix, \(\left\| . \right\| \) will denote both the euclidean norm on \({\mathbb {R}}^3\) and the operator norm on the space of k-linear maps \(L^k({\mathbb {R}}^3,{\mathbb {R}}^3)\) associated to it. When there is no ambiguity on the domain of definition \(U\subset {\mathbb {R}}^3\) of a \(C^m\) map \(f:U\rightarrow {\mathbb {R}}^3\) (resp. \(U\rightarrow L^k({\mathbb {R}}^3,{\mathbb {R}}^3)\)), we will write

$$\begin{aligned} \left\| f \right\| _0=\sup _{p\in U}\left\| f(p) \right\| \quad \in [0,+\infty ] \end{aligned}$$

and

$$\begin{aligned} \left\| f \right\| _m=\max _{1\le k\le m }\left\| D^kf \right\| _0. \end{aligned}$$
(1)

Reduction to open sets of euclidian space

The statement we will need in \({\mathbb {R}}^3\) is the following (cf. Fig. 16).

Lemma 6.2

Let U be an open subset of \({\mathbb {R}}^3,\,F\) a closed subset of U and \(\xi \) a K-plane field on U integrable on \((K \times \mathop {\mathrm {Op}}\nolimits F) \cup (L \times U)\). Given a compact subset \(A \subset U\), there exists a K-plane field \( {\bar{\xi }} \) on U satisfying the following properties:

  1. 1.

    there is a compactly supported homotopy from \(\xi \) to \( {\bar{\xi }} \) relative to \((K \times \mathop {\mathrm {Op}}\nolimits F) \cup (L \times U)\);

  2. 2.

    \( {\bar{\xi }} \) is integrable on \(K \times (A_* {\setminus } B)\) and almost horizontal on \(K \times B\), where \(A_*\) is a compact neighborhood of A and B a collection of balls in \(\mathop {\mathrm {Int}}A_* {\setminus } F\).

Fig. 16
figure16

Setup in Lemma 6.2

Proof of Proposition 6.1 assuming Lemma 6.2

Let \(M,\,B\) and \(\xi \) be as in Proposition 6.1 and \(A_{0*}\) be a compact neighborhood of B such that \(\xi \) is integrable on \(K \times (\mathop {\mathrm {Op}}\nolimits A_{0*}{\setminus } B)\). Consider open charts \(V_i \subset M,\,1 \le i \le p\), and compact subsets \(W_i \subset V_i\) such that \(M = \bigcup W_i\). Lemma 6.2 applied to

$$\begin{aligned} U_1 = V_1{\setminus } B,\quad F_1 = U_1 \cap A_{0*}, \quad A_1 = W_1 {\setminus } \mathop {\mathrm {Int}}A_{0*} \end{aligned}$$

and to the K-plane field \(\xi \) restricted to \(U_1\), provides a compact set \(A_{1*}\), a collection of balls \(B_1 \subset \mathop {\mathrm {Int}}A_{1*}\) and a new K-plane field \(\xi _1\) on \(U_1\) equal to \(\xi \) on \((K\times \mathop {\mathrm {Op}}\nolimits F_1)\cup (L\times U_1)\), and which extends to M by \(\xi _1 = \xi \) on \(M{\setminus } U_1\). We then apply Lemma 6.2 to

$$\begin{aligned} U_2 = V_2{\setminus } (B \cup B_1),\quad F_2 = U_2 \cap (A_{0*} \cup A_{1*}), \quad A_2 = W_2{\setminus }\mathop {\mathrm {Int}}(A_{0*} \cup A_{1*}) \end{aligned}$$

and to the K-plane field \(\xi _1\) restricted to \(U_2\). We then iterate this construction and after finitely many steps we are done. \(\square \)

Almost horizontality and curvature

The following lemma will be used to make sure that the plane fields we construct in the next sections have the desired almost horizontality property. It corresponds to Lemmas 2.4.1 and 2.4.2 in [8] (stated without proof in [8] and proved in [10], cf. 4.7.17 and 4.7.18). Figure 17 below gives a schematic picture of its content.

Fig. 17
figure17

2-dimensional schematic picture for Lemma 6.3

Let \(\xi \) be a transversely oriented plane field on an open subset U of \({\mathbb {R}}^3\). For every \(p \in U\), we denote by \(\xi ^+(p)\) the open half-space of \(T_p{\mathbb {R}}^3\) lying on the positive side of \(\xi (p)\) and by \(\xi ^\perp (p) \in \xi ^+(p)\) the positive unit normal vector. In other words, \(\xi ^\perp :U \rightarrow {\mathbb {S}}^2\) is the Gauss map of \(\xi \). For every integer \(m \ge 1\), we define the \(C^m\) norm of \(\xi \) to be the \(C^m\) norm (as defined in (1)) of its Gauss map, and denote it simply by \(\left\| \xi \right\| _m\):

$$\begin{aligned} \left\| \xi \right\| _m:=\left\| \xi ^\perp \right\| _m. \end{aligned}$$
(2)

Now given two points \(p, q \in U\), the affine planes \(P_p\) and \(P_q\) tangent to \(\xi (p)\) and \(\xi (q)\) respectively, determine a pencil, namely the set of planes containing the straight line \(P_p \cap P_q\), called the axis of the pencil. Note that this axis can be at infinity, in which case the planes of the pencil are all parallel.

Lemma 6.3

Let U be an open subset of \({\mathbb {R}}^3,\,\xi \) a \({C}^1\)-bounded plane field on U and \(S_* \subset {\mathbb {R}}^3\) a strictly convex sphere. For \(d_0 > 0\) sufficiently small, every image \(S \subset U\) of \(S_*\) by a dilation by a factor \(d \le d_0\) has the following properties:

  1. 1.

    \(\xi \) is tangent to S at exactly two points, a north pole \(p_+\) where their coorientations coincide and a south pole \(p_-\) where they are opposite; we denote by \(\eta \) the distribution of tangent planes to the pencil defined by \(\xi _{p_-}\) and \(\xi _{p_+}\) (the coorientation of \(\xi \) naturally endows \(\eta \) with a coorientation);

  2. 2.

    For every \(\varepsilon >0\), there exists a nonsingular vector field \(\nu \) on the ball B bounded by S, which lies in the dihedral cone \(\Omega _p = \xi _p^+ \cap \eta _p^+\) at every \(p\in B\), and which is tangent to S outside the \(\varepsilon \)-neighborhood of the poles.

Proof

Let \(c = \left\| \xi \right\| _1\) and let \(k>0\) be a (uniform) lower bound on the principal curvatures of \(S_*\)—so the principal curvatures of S are everywhere at least k / d.

Let \(\gamma :S \rightarrow {\mathbb {S}}^2\) be the Gauss map of S. The curvature hypothesis means that \(\gamma \) is a diffeomorphism and that its inverse satisfies \(\left\| D\gamma ^{-1} \right\| _0 \le d/k\). Thus,

$$\begin{aligned} \left\| D (\xi ^\perp \circ \gamma ^{-1}) \right\| _0\le \left\| D\xi ^\perp \right\| _0\left\| D\gamma ^{-1} \right\| _0 \le cd/k. \end{aligned}$$

For all \(d < k/c\), the maps \(\pm \xi ^\perp \circ \gamma ^{-1} :{\mathbb {S}}^2 \rightarrow {\mathbb {S}}^2\) are contractions and each of them has a unique fixed point denoted by \(\gamma (p_\pm )\). The points \(p_\pm \) are the poles we are looking for. As for the vector field \(\nu \), it is easily obtained with a partition of unity, provided \(\Omega _p\) (resp. \(T_pS \cap \Omega _p\)) is nonempty for every p in B (resp. in \(S{\setminus }\{p_-,p_+\}\)).

Let \(p\in B\). Clearly, the angle between \(\xi _p^\perp \) and \( \eta _p^\perp \) satisfies

$$\begin{aligned} \angle \left( \xi _p^\perp , \eta _p^\perp \right) \le \angle \left( \xi _p^\perp , \xi _{p_+}^\perp \right) + \angle \left( \xi _{p_+}^\perp , \xi _{p_-}^\perp \right) \le 2 \left\| \xi \right\| _1 d \delta _* \end{aligned}$$

where \(\delta _*\) denotes the diameter of \(S_*\). Thus, for \(d < \pi /(2c\delta _*)\), the planes \(\xi _p\) and \(-\eta _p\) are distinct, and hence \(\Omega _p\) is nonempty.

Now let \(p \in S {\setminus } \{p_-,p_+\}\). The plane \(T_pS\) is transverse to both \(\xi _p\) (by definition of \(p_\pm \)) and \(\eta _p\) (by convexity of S), and it is easy to see that \(T_pS \cap \Omega _p\) is empty if and only if \(\pm \gamma (p)\) belongs to the minimizing geodesic segment of \({\mathbb {S}}^2\) joining \(\xi _p^\perp \) to \(\eta _p^\perp \). Here we will discuss the case of \(\gamma (p)\); for \(-\gamma (p)\), replace \(p_+\) by \(p_-\).

Let \(\rho \) be the distance in B between p and \(p_+\). On \({\mathbb {S}}^2\), the disk D of radius \(c\rho \) centered at \(\xi _{p_+}^\perp \) contains \(\xi _p^\perp \) but not \(\gamma (p)\) if \(d < k/c\), for the principal curvatures of S are then greater than c. Moreover, since \(d < \pi / (2c\delta _*)\), the disk D is geodesically convex: \(c\rho \le c d \delta _* < \pi /2\). To conclude, all we need to check is that if d is small enough, D contains \(\eta _p^\perp \). This is done below, by showing that

$$\begin{aligned} \big \Vert \eta \mathbin {|}{}_{B} \big \Vert _1 \le \kappa c \end{aligned}$$

for some constant \(\kappa \) given by the geometry of \(S_*\).

First note that the norm of \(D\eta ^\perp \) at any point p is the inverse of the distance from p to the axis A of the pencil. Actually, in euclidian coordinates in which A is the z-axis, the map \(\eta ^\perp \) is of the form

$$\begin{aligned} (x,y,z) \longmapsto (x^2 + y^2)^{-\frac{1}{2}} (-y,x,0), \end{aligned}$$

so we can calculate the differential and its norm.

Now observe that the axis A remains distant from B. This is because B contains a euclidian (round) ball \(B'\) of radius \(d r_*\), where \(r_*\) only depends on the geometry of \(S_*\). The angle of the sector of the pencil between \(P_-\) and \(P_+\) (the affine planes tangent to S at \(p_-\) and \(p_+\)) is bounded above by \(c d \delta _*\). The fact that this sector contains \(B'\) implies that the distance l from the center of \(B'\) to A satisfies \(d r_* / l \le \sin (c d \delta _* / 2)\). The desired estimate follows, provided d is sufficiently small. \(\square \)

Triangulation and Key Lemma

The following result, combined with Lemma 6.3, is the key to Lemma (cf. Introduction of the Appendix) and is an adaptation of Lemma 2.3.4 in [8]. It involves a specific triangulation \(\Delta \) of \({\mathbb {R}}^3\) defined as follows.

The unit cube \([0,1]^3 \subset {\mathbb {R}}^3\) decomposes into six tetrahedra intersecting along the diagonal from (0, 0, 0) to (1, 1, 1). This subdivision of the cube gives rise to an infinite triangulation of \({\mathbb {R}}^3\) invariant under \({\mathbb {Z}}^3\), sometimes called crystalline, whose vertices are the integer points. We take the first barycentric subdivision of this triangulation (whose simplices have a diameter less than or equal to \(\sqrt{3}/2\)) and, as in Thurston’s “Jiggling Lemma” [22], we “jiggle” it in a \((2{\mathbb {Z}}^3)\)-periodic way so that any three edges sharing a vertex have linearly independent directions. One can make the jiggling small enough that the diameters of the simplices remain less than 1. We denote by \(\Delta \) the resulting triangulation. By periodicity, the distance between disjoint simplices of \(\Delta \) is bounded below by some positive number \(\delta >0\). For any \(d>0\), we denote by \(d\Delta \) the image of \(\Delta \) under a dilation by a factor d. This way, we obtain arbitrarily fine triangulations of \({\mathbb {R}}^3\) whose 3-simplices are all small (similar) copies of a finite number of model simplices.

We denote by \(N_\varepsilon (V),\,\varepsilon > 0\), the (closed) \(\varepsilon \)-neighborhood of a subset V of \({\mathbb {R}}^3\).

Key Lemma 6.4

Let U be an open subset of \({\mathbb {R}}^3,\,F\) a closed subset of U and \(\xi \) a K-plane field on U which is integrable on \((K \times \mathop {\mathrm {Op}}\nolimits F) \cup (L \times U)\). Given a compact subset \(A \subset U\), one can find positive numbers \(d_*,\,\mu \) and c such that, for every \(d<d_*\), there exists a K-plane field \( {\bar{\xi }} \) on U with the following properties:

  1. 1.

    there is a compactly supported homotopy from \(\xi \) to \( {\bar{\xi }} \) relative to \((K \times \mathop {\mathrm {Op}}\nolimits N_d(F)) \cup (L \times U)\);

  2. 2.

    \( {\bar{\xi }} \) is integrable on \(K \times N_{\mu d}(A_d^2)\) where \(A_d\) is a compact polyhedral neighborhood of A in \(d \Delta \) and \(A_d^2\) is the 2-skeleton of \(A_d\);

  3. 3.

    \(\left\| {\bar{\xi }} _t \mathbin {|}{}_{ N_{\mu d} (A_d)} \right\| _1 \le c\) for all \(t \in K\).

Proof of Lemma 6.2 assuming Lemma 6.4

Let \(U,\,F,\,\xi \) and A be as in Lemma 6.2, and let \(d_*,\,\mu \) and c be the positive numbers given by Lemma 6.4. Denote by \(\sigma _i,\,1 \le i \le p\), the model 3-simplices of the triangulation \(\Delta \). Each of them contains a strictly convex sphere \(S_i\) in the \(\mu \)-neighborhood of its boundary. For every \(d < d_*\), Lemma 6.4 provides a K-plane field \( {\bar{\xi }} \) and a polyhedral neighborhood \(A_d\) of A. Every 3-simplex \(\sigma \) of \(A_d\) contains a ball \(B_{\sigma }\) whose boundary is the image under a dilation of factor d of one of the model spheres \(S_i\). Now for every d, the plane field \( {\bar{\xi }} \) given by Lemma 6.4 satisfies \(|| {\bar{\xi }} _t \mathbin {|}{}_{ N_{\mu d} (A_d)} ||_1 \le c\) for all \(t \in K\). So according to Lemma 6.3, if d is chosen small enough (with respect to the geometry of the model spheres \(S_i\)) \( {\bar{\xi }} \) is almost horizontal on every ball \(B_{\sigma }\). If \(G\subset U{\setminus } N_d(F)\) denotes a compact subset such that the support of the deformation from \(\xi \) to \( {\bar{\xi }} \) is contained in \(K\times G\), the K-plane field \( {\bar{\xi }} \), the neighborhood \(A_* = A_d\) of A and the collection of balls B made of the \(B_\sigma \) meeting \(A_d\cap G\) (so that \(B \subset (\mathop {\mathrm {Int}}A_*) {\setminus } F\)) satisfy all the properties of Lemma 6.2. \(\square \)

Lemma 6.4 is by far the most technical result in this appendix. Its proof takes up the next two subsections.

Deformation model

Lemma 6.6 and its proof describe the properties of the deformation model we will use in the next subsection to make plane fields integrable in a neighborhood of each simplex of the 2-skeleton of some subcomplex of the triangulation \(d\Delta \), for some small enough d. More precisely, the model will be applied after rescaling the plane fields by a factor 1 / d, so our model here deals with plane fields defined near a simplex \(\sigma \) of the “big” triangulation \(\Delta \); no scaling factor d is involved in this subsection.

Our construction is directly inspired by that of Eliashberg in Lemma 2.3.2 of [8], and simply consists in flowing the restriction of the given plane field to some transverse surface under a flow tangent to the plane field, whose orbits cover a neighborhood of the simplex \(\sigma \) (cf. first paragraph of the proof below). (In other words, here, we untwist the plane field around a line field tangent to the plane field, while Eliashberg twists it to make it contact). But, as we explained in the introduction to the appendix, in the next subsection (proof of the Key Lemma), we will also need a bound on the \(C^1\) norm of the resulting plane field in terms of the geometric setting and of the norm of the initial plane field (but not of the plane field itself), and this is actually the main issue of this subsection:

Remark 6.5

Despite Eliashberg’s claim in [8, Note 2.3.3], the \({C}^1\) norm of the plane field \(\xi ^1\) given by our deformation model is not controlled by the \({C}^1\) norm of the initial plane field \(\xi \) but only by its \({C}^2\) norm. More generally, the \({C}^m\) norm of \(\xi ^1\) is controlled by the \({C}^{m+1}\) norm of \(\xi \). This “consumption” of one derivative, which comes from the “pull-back” construction of \(\xi ^1\), complicates the statement and proof of Lemma 6.6 and its application in the next subsection but does not affect the result: though the number of simplices of \(d\Delta \), and thus the number of times one applies the deformation model, grows with d, the model is actually applied simultaneously to many simplices, in a finite number of steps (independent of d), so knowing that the initial plane field is \({C}^m\)-bounded with m sufficiently large (independent of d) will give a \({C}^1\) bound on the final plane field regardless of the chosen scaling factor.

The statement of Lemma 6.6 is already quite elaborate so let us introduce part of the setting beforehand. In this subsection, we work in \({\mathbb {R}}^3\) endowed with the triangulation \(\Delta \) and with an affine orthonormal frame (Oxyz) which is not necessarily the canonical one. We denote by V the \(\delta /2\)-neighborhood of a simplex \(\sigma \) of \(\Delta \), where \(\delta \) is the minimal distance between two disjoint simplices of \(\Delta \). We endow V with the horizontal foliation \(\eta \) defined by \(dz=0\), and the plane fields \(\xi \) we deform below satisfy the following condition:

(\(*\)):

the angle between the vectors \(\xi ^\perp \) and \(\partial _x\) is everywhere less than some fixed number \(\tilde{\theta }\in (0, \pi /2)\).

In particular, \(\xi \) is transverse to \(\partial _x\), and a fortiori to \(\eta \), and the angle between the line field \(\xi \cap \eta \) and \(\partial _y\) is everywhere less than \(\tilde{\theta }\).

Given a plane field \(\xi \), all the deformations of \(\xi \) we will define consist in “straightening” \(\xi \) by (un)rotating it around \(\xi \cap \eta \) and have compact support in \(\mathop {\mathrm {Int}}V\). We will thus refer to a plane field as admissible if it contains \(\xi \cap \eta \) and coincides with \(\xi \) near the boundary \(\partial V\).

Lemma 6.6

Let \(\xi \) be a plane field on V satisfying Condition \(({*})\) and \(\left\| \xi \right\| _1<1\), and let S be a properly embedded surface in V. Given positive numbers \(\mu \) and \(\kappa \), assume \(S{\setminus }\partial S\) contains a disk D in V transverse to \(\xi \cap \eta \) whose orbit segments under \(\xi \cap \eta \) cover the \(2 \mu \)-neighborhood of \(\sigma \) and whose intersection \(D \cap P\) with any leaf P of \(\eta \) is a connected curve whose angle with \(\xi \cap \eta \) is greater than \(\kappa > 0\) (Fig. 18). Then one can deform \(\xi = \xi ^0\) by a homotopy \(\xi ^u,\,u \in [0,1]\), of admissible plane fields satisfying the following properties:

  • \(\xi ^1\) coincides with \(\xi \) along D and is integrable on the \(\mu \)-neighborhood of \(\sigma \);

  • \(|| \xi ^u ||_{m} \le \chi _m \bigl ( || \xi ||_{m+1} \bigr )\) for all \(u \in [0,1]\) and all \(m \ge 1\), where \(\chi _m\) is a polynomial without constant term and with positive coefficients depending only on \(\tilde{\theta },\,\kappa ,\,\mu \) and S (but not on D).

Moreover, the homotopy \(\xi ^u\) varies continuously with \(\xi \).

Fig. 18
figure18

Two situations to which Lemma 6.6 will be applied (near special faces and non-special faces respectively). Each picture represents the intersections of the objects of Lemma 6.6 with a leaf P of the horizontal foliation \(\eta \)

Remark 6.7

The role of S is unclear at that stage. The point is that we are going to apply Lemma 6.6 to a family of plane fields \(\xi _t\), using a different \(D_t\) for each t, but all these disks will be part of the same surface S, so the bound in the second item will be uniform in t, since \(\chi _m\) depends only on S and not on D.

Remark 6.8

What really matters to us concerning \(\chi _m\) is that it is nondecreasing and that \(\chi _m(x)/x\) is bounded on any bounded set of \({\mathbb {R}}_+^*\).

Proof

Let us first describe \(\xi ^u\) geometrically. By condition \((*)\), there exists a nonvanishing vector field \(\nu \) tangent to \(\xi \cap \eta \) (and a unique one if we impose the additional condition \(\nu \cdot \partial _y = 1\)). Let C denote the flow “cylinder” of D under \(\nu \). Note that an integral curve of \(\nu \) starting from D cannot return to D. Indeed, otherwise, some subarc of this curve intersects D exactly at its endpoints, which necessarily belong to the same leaf P of \(\eta \) and are thus connected by an arc of \(P\cap D\) by assumption. Then the union of these two arcs forms a simple closed curve in \(P\cap V\) (which is convex) and thus bounds a disk in \(P\cap V\), which is either positively or negatively stable under \(\nu \). Then by a corollary of Poincaré-Bendixson’s Theorem, this disk must contain a singularity of \(\nu \), which is impossible.

Hence, C is an interval bundle over D. Now the key observation is that there is a unique integrable plane field \( {\bar{\xi }} \) on C containing \(\xi \cap \eta \) and coinciding with \(\xi \) at every point of D: the unique plane field invariant under the holonomy of \(\xi \cap \eta \) and equal to \(\xi \) along D. For \(\xi ^1\), we will take a plane field coinciding with \( {\bar{\xi }} \) on the \(\mu \)-neighborhood of \(\sigma \), with \(\xi \) outside a \(2\mu \)-neighborhood of \(\sigma \) (which is contained in C) and with both on D, and \(\xi ^u\) will be a linear homotopy connecting \(\xi \) to \(\xi ^1\).

Let us now briefly explain why the control on derivatives of the second part of the statement is natural. One can easily believe that the \(C^m\) norm of \(\xi ^u\), as “interpolation” between \(\xi \) and \( {\bar{\xi }} \) in a fixed region (\(N_{2\mu }(\sigma ){\setminus } N_\mu (\sigma )\)), is bounded in terms of the \(C^m\) norms of these two plane fields, so the main object to control is in fact \( {\bar{\xi }} \). Now \( {\bar{\xi }} \) is obtained from \(\xi \) by some kind of pull-back/push-forward construction as follows. Denote by \(\nu \) the vector field spanning \(\xi \cap \eta \) and satisfying \(\nu \cdot \partial _y = 1\), by \(\phi :\Omega \subset {\mathbb {R}}\times V \rightarrow V\) its flow, and by \(\tau :C \rightarrow {\mathbb {R}}\) the map uniquely determined by:

$$\begin{aligned} \forall p\in C, \;\phi ^{-\tau (p)}(p)\in D \end{aligned}$$

(observe that since \(\sigma \) is of diameter less than 1, and \(\delta <1\), the condition \(dy(\nu ) = 1\) implies that \(\Omega \) is contained in \([-2,2]\times V\) and that the function \(|\tau |\) is bounded by 2). Then \( {\bar{\xi }} \) is defined by:

$$\begin{aligned} {\bar{\xi }} (p) = \left( \left( \phi ^{\tau (p)}\right) _*\xi \right) (p) = D\phi ^{\tau (p)}\left( \phi ^{-\tau (p)}(p)\right) \cdot \xi \left( \phi ^{-\tau (p)}(p)\right) . \end{aligned}$$
(3)

Thus, in short, by composition, the derivatives of \( {\bar{\xi }} \) are polynomials in the derivatives of the involved objects, i.e. \(\xi \), but also \(\tau \) and the flow of \(\nu \), whose \(C^m\) norms, as we will see, are controlled by that of \(\nu \), which are in turn controlled by that of \(\xi \). As a result, since the expression of \( {\bar{\xi }} \) involves the differential of the flow, the \(C^m\) norm of \( {\bar{\xi }} \) will be controlled by the \(C^{m+1}\) norm of \(\xi \).

The uniform polynomial bound on the variations of the flow \(\phi \) in terms of its generating vector field \(\nu \) (cf. Claims 6.11 and 6.12) is rather natural; the proofs, which we only include for the sake of completeness, rely on Gronwall’s Lemma and on general formulas (cf. (6) below, for example) for the derivatives of composed maps of several variables (generalizing the so-called Faà di Bruno Formula for maps of one variable). The uniformity of the control of \(\nu \) and \(\tau \) on the other hand (cf. Claims 6.9 and 6.13) depends in a crucial way on the angle bounds \(\tilde{\theta }\) and \(\kappa \) and on the geometry of S. The calculations carried out in the respective proofs, also based on Faà di Bruno-like formulas, are only intended to clarify this dependency.

Let us now make the previous paragraph more precise.

First of all, we defined the \(C^m\) norm of a plane field as the \(C^m\) norm of its Gauss map, so here, rigorously speaking, the derivatives we want to control are those of \( {\bar{\xi }} ^\perp \), which, from (3), has the following expression:

$$\begin{aligned} {\bar{\xi }} ^\perp (p)=\frac{{}^t\!\!\left( D\phi _p^{-\tau (p)}\right) \cdot \xi ^\perp \left( \phi ^{-\tau (p)}(p)\right) }{\left\| {}^t\!\!\left( D\phi _p^{-\tau (p)}\right) \cdot \xi ^\perp \left( \phi ^{-\tau (p)}(p)\right) \right\| }. \end{aligned}$$

Let us denote the numerator by X(p) and write \(Y(t,p)={}^t(D\phi _p^{t})\cdot \xi ^\perp (\phi ^t(p))\), so that \(X(p)=Y(-\tau (p),p)\). We are going to control the variations of \(\nu \), then its flow \(\phi ^t\), then \(Y,\,\tau ,\,X,\, {\bar{\xi }} ^\perp =\frac{X}{\left\| X \right\| }\) and finally \((\xi ^u)^\perp \).

Throughout the calculations, given \(m\in {\mathbb {N}}\), the symbol \(\chi _m\) denotes some universal polynomial with positive coefficients depending, as in the lemma, only on \(m,\,\tilde{\theta },\,\mu ,\,\kappa \) and S, and which will change in the course of the argument. When we write \(\chi _m^0\) rather than \(\chi _m\), we mean that, in addition, \(\chi _m\) has no constant term. \(\square \)

Claim 6.9

For all \(m \ge 1\),

$$\begin{aligned} || \nu ||_{m} \le \chi _m^0 \bigl (|| \xi ||_{m}\bigr ). \end{aligned}$$

Remark 6.10

For \(m=1\), with the assumption \(\left\| \xi \right\| _1<1\), this implies: \(\left\| \nu \right\| _1\le c\left\| \xi \right\| _1\), for some constant c independent of \(\xi \).

Proof

If \(\xi ^\perp =u\partial _x+v\partial _y+w\partial _z\), the maps uvw satisfy \(u^2+v^2+w^2=1\) and \(\frac{v^2+w^2}{u^2}<\tan ^2\tilde{\theta }\) by condition \((*)\), so \(\frac{1}{u^2}=1+\frac{v^2+w^2}{u^2}<1+\tan ^2\tilde{\theta }\). Now \(\nu =-\frac{v}{u} \partial _x+\partial _y\), so its \(C^m\) norm is that of \(\frac{v}{u}\). Now the m-th derivative of \(\frac{v}{u}\) is a fraction with numerator a (universal) polynomial in the derivatives of v and u of order \(k\in \llbracket 0,m\rrbracket \), each monomial containing a derivative of order at least \(1,\,u\) and v are bounded above by 1, and the denominator is a power of u, for which we have a lower bound independent of \(\xi \). Hence the required polynomial bound on \(\left\| \nu \right\| _m\). \(\square \)

This leads to a similar bound on the (space) differential of the flow of \(\nu \):

Claim 6.11

\( || \phi ^t ||_1 = ||D \phi ^t ||_0 \le \chi _1 (|| \xi ||_{1}) \), and for all \(m \ge 2\), and all \(|t| \le 2\),

$$\begin{aligned} ||D^m \phi ^t ||_0 \le \chi _m^0 \bigl (|| \xi ||_{m} \bigr ) \end{aligned}$$

(with \(\chi _m\) independent of t).

To prove this, we will use the following version of Gronwall’s Lemma:

Gronwall’s Lemma

Let \((E,\left\| \cdot \right\| )\) be a finite dimensional normed vector space and \(x:I\rightarrow E\) a differentiable curve satisfying for some positive a and b

$$\begin{aligned} \forall t\in I, \left\| x'(t) \right\| < a\left\| x(t) \right\| +b. \end{aligned}$$

Then for all \(t_0\) and t in I,

$$\begin{aligned} \left\| x(t) \right\| \le \left\| x(t_0) \right\| e^{a|t-t_0|}+\frac{b}{a} \left( e^{a|t-t_0|}-1\right) . \end{aligned}$$

Proof of Claim 6.11

We proceed by induction on m. For \(m=1\), the differential \(D\phi ^t(p)\) at any point p satisfies the differential equation

$$\begin{aligned} \frac{d}{dt} D\phi ^t(p) = D\nu \bigl (\phi ^t(p)\bigr ) \, D\phi ^t(p) \end{aligned}$$
(4)

with initial condition \(D\phi ^0(p) = {\mathrm {id}}\). In particular,

$$\begin{aligned} \left\| D\phi ^0(p) \right\| = 1\quad \text {and}\quad \left\| \frac{d}{dt} D\phi ^t(p) \right\| \le \left\| \nu \right\| _1 \, \left\| D\phi ^t(p) \right\| \end{aligned}$$

so if \(\left\| \nu \right\| _1>0\), by Gronwall’s Lemma with \(t_0=0\) and \(a=b=\left\| \nu \right\| _1\) we have for all \(|t|\le 2\) that

$$\begin{aligned} \left\| D\phi ^t(p) \right\| \le 2e^{2\Vert \nu \Vert _1}-1. \end{aligned}$$

Let c be the constant given by Remark 6.10 and C such that \(e^x-1\le Cx\) for all \(x\le 2c\). Since \(2\left\| \nu \right\| _1\le 2c\left\| \xi \right\| _1\le 2c\),

$$\begin{aligned} 2e^{2\Vert \nu \Vert _1}-1 = 1+2(e^{2\Vert \nu \Vert _1}-1)\le 1+4C\Vert \nu \Vert _1\le 1+4Cc\Vert \xi \Vert _1, \end{aligned}$$

which gives the desired bound on \(\left\| D\phi ^t \right\| _0\) (if \(\left\| \nu \right\| _1=0,\,\left\| D\phi ^t(p) \right\| \equiv 1\)).

Now let \(m\ge 2\) and assume Claim 6.11 has been proved for every \(k\le m-1\). For all (tp) where it makes sense,

$$\begin{aligned} \frac{d}{dt} D^{m}\phi ^t(p) = D^m(\nu \circ \phi ^t)(p) \end{aligned}$$
(5)

with initial condition \(D^{m}\phi ^0(p) = 0\). Given a smooth function f on an open subset of \({\mathbb {R}}^3\) and a multi-index \(I=(i_1,\ldots ,i_k)\in \{1,2,3\}^k,\,1\le k \le m\), we denote by \(\partial _If\) the partial derivative \(\frac{\partial ^kf}{\partial x_{i_1}\ldots \partial x_{i_k}}\). Given a subvector \(J=(i_{n_1},\ldots ,i_{n_l})\) of I, i.e. given a subset \(B=\{n_1,\ldots ,n_l\}\) of \(\{1,\ldots ,k\}\) with \(n_1<\cdots <n_l\), we will abusively write \(\partial _Bf\) instead of \(\partial _Jf\). By induction, one gets the following formula for partial derivatives of the composed map \(\nu \circ \phi ^t\):

$$\begin{aligned} \partial _I(\nu \circ \phi ^t)(p) = \sum _{\pi \in \Pi _k}D^{|\pi |}\nu (\phi ^t(p))\cdot \left( \prod _{B\in \pi }\partial _B\phi ^t(p)\right) \end{aligned}$$
(6)

where \(\Pi _k\) denotes the set of partitions \(\pi \) of \(\{1,\ldots ,k\}\) (recall k is the length of I here), and \(|\pi |\) the number of “blocks” of such a partition. If the blocks of \(\pi \) are \(B_1\),..., \(B_k\), the parenthesis \(\left( \prod _{B\in \pi }\partial _B\phi ^t(p)\right) \) must be understood as the k-tuple of vectors \(\partial _{B_1}\phi ^t(p)\),..., \(\partial _{B_k}\phi ^t(p)\) to which \(D^{k}\nu (\phi ^t(p))\) is applied. If I is of size m, isolating the partition \(\pi \) with one block of size m, we get:

$$\begin{aligned} \partial _I(\nu \circ \phi ^t)(p)= & {} D\nu (\phi (p))\cdot \partial _I\phi ^t(p)\\&+\sum _{\mathop {\pi \in \Pi _m}\limits _{|\pi |\ge 2}} D^{|\pi |}\nu (\phi ^t(p))\cdot \left( \prod _{B\in \pi }\partial _B\phi ^t(p)\right) \end{aligned}$$

and thus

$$\begin{aligned} \left\| \partial _I(\nu \circ \phi ^t)(p) \right\| \le \Vert \nu \Vert _{1} \left\| \partial _I\phi ^t(p) \right\| + \sum _{\mathop {\pi \in \Pi _m}\limits _{|\pi |\ge 2}}\Vert \nu \Vert _{|\pi |}\prod _{B\in \pi }\left\| D^{|B|}\phi ^t \right\| \end{aligned}$$

where the last term is a \(\chi _m^0(\Vert \xi \Vert _{m})\) by induction and Claim 6.9. So according to (5),

$$\begin{aligned} \left\| \frac{d}{dt} \partial _I\phi ^t(p) \right\| \le \Vert \nu \Vert _{1} \left\| \partial _I\phi ^t(p) \right\| + \chi _m^0(\Vert \xi \Vert _{m}) \end{aligned}$$

and once again we conclude using Gronwall’s Lemma (and the fact that \(\partial _I\phi ^0(p)=0\) for all p). \(\square \)

Recall that Y is defined by \(Y(t,p)={}^t(D\phi ^t(p))\cdot \xi ^\perp (\phi ^t(p))\).

Claim 6.12

For all \(m\ge 1,\,0\le k\le m\), and every multi-index \(I=(i_1,\ldots ,i_{m-k})\in \{1,2,3\}^{m-k}\),

$$\begin{aligned} \left\| \partial _I(\partial _t)^k Y \right\| _0 \le \chi _m^0 \bigl (|| \xi ||_{m+1} \bigr ) . \end{aligned}$$

Proof

This follows easily from Claims 6.9 and 6.11, by product and composition. Let us explain how, starting with the time derivatives:

$$\begin{aligned} \partial _t Y(t,.)&= \partial _t\,{}^t(D\phi ^t)\cdot ( \xi ^\perp \circ \phi ^t) + \,{}^t(D\phi ^t)\cdot \partial _t( \xi ^\perp \circ \phi ^t)\\&={}^t(D\phi ^t) ({}^tD\nu \circ \phi ^t)\cdot ( \xi ^\perp \circ \phi ^t)+ {}^t(D\phi ^t)\cdot (D \xi ^\perp \circ \phi ^t)\cdot (\nu \circ \phi ^t)\\&={}^t(D\phi ^t)({}^tD\nu \cdot \xi ^\perp + D \xi ^\perp \cdot \nu )\circ \phi ^t\\&={}^t(D\phi ^t)(\nu * \xi ^\perp )\circ \phi ^t \end{aligned}$$

where \(\nu *\) denotes the Lie derivative-like differential operator \(X\mapsto {}^tD\nu \cdot X+ D X \cdot \nu \). By induction,

$$\begin{aligned} (\partial _t)^k Y(t,.) ={}^t(D\phi ^t)\cdot ((\nu *)^k \xi ^\perp )\circ \phi ^t \end{aligned}$$

There is a general polynomial formula for \((\nu *)^k \xi ^\perp \) in terms of the derivatives of \(\nu \) and \(\xi ^\perp \) of order \(l\in \llbracket 0,k\rrbracket \), each monomial containing a real derivative (i.e. of order at least one). Now Formula (6) applied to \((\nu *)^k \xi ^\perp \) instead of \(\nu \) gives a polynomial expression for any partial derivative of order l of \(((\nu *)^k \xi ^\perp )\circ \phi ^t\) in terms of that of \((\nu *)^k \xi ^\perp \) and \(\phi ^t\) of order \(\le l\), so in terms of the derivatives of \(\nu \) and \(\xi ^\perp \) of order \(\le k+l\) and that of \(\phi ^t\) of order \(\le l\). So in the end, any partial derivative \(\partial _I(\partial _t)^k Y\) of order l of the product \((\partial _t)^k Y={}^t(D\phi ^t)\cdot ((\nu *)^k \xi ^\perp )\circ \phi ^t\) is given by a general polynomial formula in terms of the derivatives of \(\nu \) and \(\xi ^\perp \) of order \(\le k+l\) and that of \(\phi ^t\) of order \(\le l+1\) (again, each monomial containing a real derivative of \(\nu \) or \(\xi ^\perp \)), and we conclude using Claims 6.9 and 6.11. \(\square \)

Claim 6.13

For every \(m\ge 1\) and every multi-index \(I=(i_1,\dots ,i_{m})\in \{1,2,3\}^{m}\),

$$\begin{aligned} \left\| \partial _I\tau \right\| _0 \le \chi _m \bigl (|| \xi ||_{m+1} \bigr ). \end{aligned}$$

Here \(\left\| \partial _I\tau \right\| _0\) simply means \(\sup _{p\in V}|\partial _I\tau (p)|\).

Proof

Since \(\tau \circ \phi ^t = \tau + t\), given Claim 6.11, we only need to estimate the derivatives of \(\tau \) along D. To that end, we need to understand the relation between \(\tau \) and the geometry of D.

Therefore, let us introduce the function \(\tau _0\), defined in a neighborhood of D, whose restriction to every plane P of \(\eta \) is the euclidean distance to \(S\cap P\) multiplied by the sign of \(\tau \). In other words \(\tau _0\) is the algebraic distance to \(S\cap P\)—where S is cooriented so that \(\tau \) and \(\tau _0\) have the same sign—and is thus smooth. Moreover, it depends only on the geometric setting (not on \(\xi \)). The idea is to compare \(\tau \) to \(\tau _0\) and to deduce a bound on the first from one on the latter. To that end, we consider the unique multiple \(\nu _0=f\nu \) of \(\nu \) near D whose flow \(\phi _0^t\) satisfies:

$$\begin{aligned} \forall p, \phi _0^{-\tau _0(p)}(p)\in D. \end{aligned}$$

Such a vector field must satisfy \(\tau _0\circ \phi _0^t=\tau _0+t\) which, differentiating with respect to t, gives \(f= \frac{1}{\nu \cdot \tau _0}\). The flows \(\phi _0^t\) and \(\phi ^t\) satisfy the relation

$$\begin{aligned} \phi _0^t(p) = \phi ^{s(t,p)}(p) \end{aligned}$$

where the function s satisfies \(s(0,p) = 0\) for all p and the differential equation

$$\begin{aligned} \frac{d}{dt} s(t,p) = f \left( \phi ^{s(t,p)}(p)\right) . \end{aligned}$$
(7)

Since

$$\begin{aligned} \phi _0^{-\tau _0(p)}(p) = \phi ^{-\tau (p)}(p) = \phi ^{s(-\tau _0(p),p)}(p), \end{aligned}$$

we have \(-\tau (p) = s(-\tau _0(p),p)\). Since \(s(0,p) = 0\) for all p close to D in C, the spatial derivatives \(\partial _Is(0,p)\) are all zero. As a consequence, for \(p \in D\), the general formula (which we will omit here) expressing the derivatives of \(\tau \) in terms of that of s and \(\tau _0\) becomes

$$\begin{aligned} -\partial _I\tau (p) = \sum _{\pi \in \Pi _k} (\partial _t)^{|\pi |} s(0,p) \left( \prod _{B \in \pi } \partial _{B} \tau _0(p) \right) , \end{aligned}$$
(8)

where k is the length of the multi-index I. The quantities \(||\partial _{B} \tau _0 ||_0\) depend only on the geometry of S. We are now going to control \((\partial _t)^{k+1} s(0,.)\) by induction. For \(k=0\), according to Eq. (7), we need a uniform bound on \(f= \frac{1}{\partial _\nu \tau _0}\). But \(\partial _\nu \tau _0(p)\), for all \(p \in D\), is the scalar product of \(\nu (p)\) with the unit normal vector to \(S \cap P\) in P, where P is the horizontal plane containing p. The function \(\partial _\nu \tau _0\) is thus bounded below along D by some constant depending only on \(\kappa \) and \(\tilde{\theta }\).

Now for \(k\ge 1\), differentiating Eq. (7), one gets, for all \(p \in D\),

$$\begin{aligned} (\partial _t )^{k+1}s(0,p) = \sum _{\pi \in \Pi _k} \left( \partial _{\nu }^{|\pi |} f(p)\right) \prod _{B \in \pi } \left( \partial _t^{|B|} s(0,p)\right) , \end{aligned}$$
(9)

where \(\partial _{\nu }\) denotes the derivative in the direction of \(\nu \) (in other words, \(\partial _\nu f = \nu \cdot f\)). We saw that \(\frac{1}{\partial _\nu \tau _0}\) is bounded above independently of \(\xi \). Moreover, every quantity \(||\partial _\nu ^{l+1} \tau _0 ||_0\) is bounded above by a function of \(||\nu ||_{l}\) (which depends only on S). Thus, every quantity \(||\partial _\nu ^{|\pi |} f \mathbin {|}{}_{D} ||_0\) is itself controlled by \(||\nu ||_{|\pi |}\), and Relation (9) shows by induction that the quantities \(|\partial ^{k}_t s(0,p)|\) are controlled by \(||\nu ||_{k}\). Formula (8) and Claim 6.9 thus imply Claim 6.13. \(\square \)

Claim 6.14

For all \(m\ge 1\),

$$\begin{aligned} \left\| X \right\| _m \le \chi _m^0 \bigl (\left\| \xi \right\| _{m+1} \bigr ). \end{aligned}$$

Proof

Since \(X=Y\circ (-\tau ,{\mathrm {id}})\), this follows by composition from Claims 6.12 and 6.13. \(\square \)

Since \( {\bar{\xi }} ^\perp =X/\Vert X\Vert \), in order to deduce that \(\left\| {\bar{\xi }} \right\| _{m} \le \chi _m^0(|| \xi ||_{m+1} )\), we should still check that \(\Vert X\Vert \) is bounded below (independently of \(\xi \)). We will simply say here that this follows from the fact that \(\xi ^\perp \) is unitary and that \(D\phi _p^{-\tau (p)}\) is close enough to the identity in our setting.

Now let \(\rho : V \rightarrow [0,1]\) be a function equal to 1 on \(N_{\mu }(\sigma )\), with support in \(N_{2 \mu }(\sigma )\). For all \(u \in [0,1]\), define \(\xi ^u\) by

$$\begin{aligned} (\xi ^u)^\perp =(1 - u \rho ) \xi ^\perp + u \rho {\bar{\xi }} ^\perp . \end{aligned}$$

Observe that these vector fields are all nonsingular, for the \(\partial _x\) components of \( \xi ^\perp \) and \( {\bar{\xi }} ^\perp \) are both positive. The plane fields \(\xi ^u\) have all the desired properties. Let us simply stress that the bound on their derivatives is independent of the particular simplex \(\sigma \) under scrutiny. This is due to the \((2{\mathbb {Z}})^3\)-invariance of \(\Delta \): all simplices are copies of a finite number of model ones, and we only need one cut-off function \(\rho \) per isometry class of simplices (note that \(\sigma \) plays no role in the rest of the proof).

Remark 6.15

Note that, if the plane field \(\xi \) is already integrable on a region of the form \(C' = \{ \phi ^t(p), \; p \in D', \; t \in [a(p),b(p)] \}\) for some domain \(D' \subset D\) and some functions \(a, b :D' \rightarrow {\mathbb {R}}\) satisfying \(a \le 0 \le b\), then the homotopy \(\xi ^u\) of Lemma 6.6 is stationary on \(C'\).

Proof of the Key Lemma 6.4

We start with the data \(U,\,F,\,A\) and \(\xi \) of Lemma 6.4 and use the notations of Sect. 6.3. As explained in the introduction to the appendix, the Key Lemma is proved by induction on the successive skeleta of some triangulation of the parameter space K. This induction is formalized in Lemma 6.19 below (cf. Sect. 6.5.4), whose proof takes up the last subsection of the article. Before stating this Lemma, we need to prepare its setting, i.e.:

  • to define, for any scaling factor d (less than some \(d_0\) defined below), polyhedral neighborhoods \(A_d\) and \(F_d\) of A and F, the support of the future deformation(s) being contained in \(A_d\) and disjoint from \(F_d\) (cf. Sect. 6.5.1);

  • to fix the triangulation of the parameter space (cf. 6.5.2);

  • to define special simplices precisely (cf. 6.5.3).

Recall that \(\Delta \) is the \((2{\mathbb {Z}})^3\)-invariant triangulation of \({\mathbb {R}}^3\) obtained by “jiggling” the barycentric subdivision of the crystalline triangulation with integer vertices. By periodicity of the construction, \(\Delta \) has a finite number of model simplices, meaning that every simplex of \(\Delta \) is the image of one of those by a translation. The diameter of the simplices of \(\Delta \) is less than 1. Moreover, the distances between two disjoint simplices and the angles between two intersecting 1 or 2-simplices (not contained into one another) are uniformly bounded below by numbers denoted by \(\delta > 0\) and \(\gamma \in (0, \pi /2]\) respectively (the angle between a straight line and a plane is the angle between the straight line and its orthogonal projection on the plane).

Fix an angle \(\theta < \gamma /2\).

Polyhedral neighborhoods

Given \(d>0\), we will still (improperly) call every subcomplex coming from a cube which has been subdivided, “jiggled” and scaled, a “cube” of \(d\Delta \). Since A is a compact subset of U, for \(d_0 > 0\) sufficiently small, \(N_{2d_0}(A)\) is contained in U and \(\xi \) is integrable on \(K \times N_{2d_0} (F\cap A)\). Fix such a \(d_0\).

Given \(d < d_0/4\), we denote by \(A_d\) and \(F_d\) the subcomplexes of \(d\Delta \) made up of all the “cubes” meeting \(N_{d_0} (A)\) and \(N_{d_0} (F \cap A)\) respectively. Thus, since \(d_0 + 3d < 2d_0\),

$$\begin{aligned} N_{d_0} (A) \subset A_d&\subset N_d(A_d) \subset N_{2d_0} (A) \\ \text {and}\quad N_{d_0} (F \cap A) \subset F_d&\subset N_{d}(F_d) \subset N_{2d_0} (F \cap A) \end{aligned}$$

so, in particular, \(\xi \) is integrable on \(K \times N_{d}(F_d)\).

Remark 6.16

The combinatorial structure of \(\Delta \) will be important when it comes to perturb the plane fields near (special) simplices whose boundary meets \(F_d\), where the homotopy must be stationary. More precisely, we will need to know that every 2-simplex of \(A_d\) not contained in \(F_d\) has at most one edge in \(F_d\). Indeed, let \(\sigma \) be such a 2-simplex and Q the cube of \(A_d\) containing it. By assumption, this cube is not contained in \(F_d\), so \(\sigma \cap F_d \subset \sigma \cap \partial Q\). Since the triangulation \(\Delta \) is obtained by barycentric subdivision, there are two cases:

  • if \(\sigma \) has a vertex in the interior of Q, it has at most one edge in \(\partial Q\);

  • otherwise, \(\sigma \subset \partial Q\) has a vertex q in the interior of some “square face” of Q; either \(q \in F_d\) and then \(\sigma \subset F_d\) (for \(F_d \cap Q\) is a union of “square faces”), or \(q \notin F_d\) and then \(\sigma \) has at most one edge in \(F_d\).

Subdivision of the parameter space

We fix a subdivision of the parameter space K compatible with L and so fine that the following inequality holds on every simplex \(K_*\):

For \(0 \le i \le n\), where \(n = \dim K\), denote by \(K^i\) the union of the i-skeleton of the triangulation with the subcomplex L. We also write \(K^{-1} = L\).

Special simplices

Here, d is any positive number less than \(d_0\).

Definition 6.17

Given a simplex \(K_*\) of K and a \(K_*\)-plane field \(\xi ^*\) on U, we will call a 2-simplex \(\sigma \) of \(A_d\) special (for \(\xi ^*\)) if it is not contained in \(F_d\) and if there exists \((s, q) \in K_* \times \sigma \) such that \(\angle ( \sigma , \xi ^*_s(q) ) < \theta /2\).

Claim 6.18

If \(\xi ^*\) satisfies:

$$\begin{aligned} \forall (s,t,p)\in (K_*)^2\times U, \quad \angle (\xi ^*_s(p),\xi ^*_t(p))<\frac{\theta }{8} \end{aligned}$$
$$\begin{aligned} \mathrm{and}~~\forall (t,p,q)\in K_*\times (A_d)^2 \text { s.t. } |p-q|<2d, \quad \angle (\xi ^*_t(p),\xi ^*_t(q))<\frac{\theta }{8},\quad \end{aligned}$$

then for any special simplex \(\sigma \),

$$\begin{aligned} \angle ( \sigma , \xi ^*_t(p) ) < \theta \quad \text {for all}\, (t,p) \in K_* \times N_d (\sigma ). \end{aligned}$$

In particular, special simplices are disjoint.

Proof

If \((s,q) \in K_* \times \sigma \) is such that \(\angle ( \sigma , \xi ^*_s(q) ) < \theta /2\), then for all \((t,p) \in K_* \times N_d (\sigma )\),

$$\begin{aligned} \angle ( \sigma , \xi ^*_t(p) )&\le \angle ( \sigma , \xi ^*_s(q) ) + \angle ( \xi ^*_s(q), \xi ^*_s(p) ) + \angle ( \xi ^*_s(p), \xi ^*_t(p) ) \\&< \frac{\theta }{2} + \frac{\theta }{8} + \frac{\theta }{8} <\theta . \end{aligned}$$

Now assume that \(\sigma \) and \(\sigma '\) are non disjoint special simplices. Let \(s, s' \in K_*\) and \(q \in \sigma ,\,q' \in \sigma '\) be such that

$$\begin{aligned} \angle ( \sigma , \xi ^*_s(q) ) < \frac{\theta }{2} \quad \text {and} \quad \angle ( \sigma ', \xi ^*_{s'} (q') ) < \frac{\theta }{2}. \end{aligned}$$

Then

$$\begin{aligned} \angle ( \sigma , \sigma ')&\le \angle ( \sigma , \xi ^*_s(q) ) + \angle ( \xi ^*_s(q), \xi ^*_s(q') ) + \angle ( \xi ^*_s(q'), \xi ^*_{s'}(q') ) + \angle ( \xi ^*_{s'}(q'), \sigma ' ) \\&< \frac{\theta }{2} + \frac{\theta }{8} + \frac{\theta }{8} + \frac{\theta }{2} < 2\theta < \gamma . \end{aligned}$$

By definition of \(\gamma \), the simplices \(\sigma \) and \(\sigma '\) coincide. \(\square \)

Induction Lemma

Given a plane field \(\xi \), we denote by \(\Vert \xi \Vert _m\) the norm of its restriction to \(N_{2d_0}(A)\) and for all \(d>0\), we define

$$\begin{aligned} \left\| \xi \right\| _{d,m} =\left\| (h_d)^*\xi _{|N_{2d_0}(A)} \right\| _m \end{aligned}$$

where \(h_d\) denotes any homothety of factor d. Note that

$$\begin{aligned} ((h_d)^*\xi )^\perp = d\cdot (h_d)^*(\xi ^\perp ) \end{aligned}$$

so

$$\begin{aligned} \left\| \xi \right\| _{d,m} = d\left\| (h_d)^*(\xi ^\perp ) \right\| _m \end{aligned}$$

and in particular

$$\begin{aligned} \left\| \xi \right\| _{d,1} =d \left\| \xi \right\| _1 \quad \text {and}\quad \left\| \xi \right\| _{d,m} \le d \left\| \xi \right\| _m\quad \forall m\ge 1. \end{aligned}$$

The Key Lemma 6.4 is a consequence of the following result.

Lemma 6.19

For every \(0 \le i \le n+1\), there are positive numbers \(d_i,\,\mu _i\) and \((c_{i,m})_{m \ge 1}\) such that, for every \(d < d_i\), there exists a compactly supported homotopy \(\xi ^u,\,u \in [0,i]\), of \(K^{i-1}\)-plane fields on U with the following properties

  • \(\xi ^0\) coincides with \(\xi \) (or more accurately with its restriction to \(K^{i-1} \times U\)) and the homotopy is relative to \((K^{i-1} \times \mathop {\mathrm {Op}}\nolimits N_{\mu _id} (F_d)) \cup (L \times U)\);

  • \(\xi ^i\) is integrable on \(K^{i-1} \times N_{\mu _id} (A_d^2)\);

  • for every \((t, u) \in K^{i-1} \times [0,i]\),

  • for every \((t, u) \in K^{i-1} \times [0,i]\) and every \(p\in U\),

    $$\begin{aligned} \angle ( \xi _t^u(p), \xi _t(p) ) < \frac{\theta }{32}. \end{aligned}$$

Proof of the Key Lemma 6.4

For \(i = n+1\), the above lemma implies the Key Lemma with \(d_* = d_{n+1},\,\mu = \mu _{n+1}\), and \(c = c_{n+1,1}\). \(\square \)

Proof of Lemma 6.19

We proceed by induction. Step \(i = 0\) is trivial (with \(\mu _0 = 1\)) since the plane fields \(\xi _t,\,t \in K^{-1} = L\), are integrable on all of U and uniformly \(C^m\)-bounded on \(N_{2d_0}(A)\) for every m.

Assume now that step \(i \ge 0\) has been completed and let us describe the structure of step \(i+1\) (each of the following sentences will be detailed afterwards). First, given any \(d<d_i\), we take the homotopy of \(K^{i-1}\)-plane fields given by step i and extend it to a homotopy of \(K^i\)-plane fields stationary outside a neighborhood of \(K^{i-1}\). Taking \(d_i\) smaller if necessary, we arrange that for any \(d<d_i\), the restriction of the resulting \(\xi ^i\) to any simplex \(K_*\) of \(K^i\) satisfies the hypothesis of Claim 6.18, so that, for any \(K_*\), special simplices are disjoint.

Then we build a homotopy \(\xi ^{i+u},\,u \in [0,1]\), of \(K^i\)-plane fields relative to \(K^{i-1} \times U\). This is done i-simplex by i-simplex of \(K^i\) independently, applying Lemma 6.6 to a neighborhood of each simplex of the 2-skeleton of \(A_d\) not contained in \(F_d\), after a rescaling by a factor 1 / d, and taking these simplices in a suitable order: first special simplices, then vertices contained in no such simplex, then edges, and finally non-special faces. For each i-simplex of \(K^i\), each of these four substeps yields “one quarter” of the desired homotopy from \(\xi ^i\) to \(\xi ^{i+1}\), namely \(\xi ^{i+u},\,u \in [0,1/4],\,[1/4,1/2],\,[1/2,3/4]\), and [3 / 4, 1].

More precisely, these four substeps follow the same pattern: we first define coordinates, constants (\(\mu ,\,\kappa \)) and surfaces (\(S,\,D\)) to which Lemma 6.6 and Remark 6.15 (to make the deformation relative to the already foliated region) are applicable (provided d is small enough). Then (taking d even smaller if necessary), we use the quantitative part of Lemma 6.6 to bound the variations of the resulting plane fields (in space and time), to ensure both the applicability of the deformation model in the next substep and the disjointness of special simplices in the next inductive step (\(i+2\)). In particular, the set of d’s for which the construction can actually be carried on decreases at each substep.

This similarity of pattern makes this last subsection somewhat repetitive, and generates references to a large number of very similar inequalities. The cases of special simplices and of vertices are probably enough for the reader to get the whole picture. There are no new ideas in the rest; we mainly include it for the sake of completeness and to show how Remark 6.15 is used in each particular case to guarantee the global coherence of the construction.

Before detailing these four substeps, let us be a little bit more specific about the \(K^i\)-plane field \(\xi ^i\) defined after step i. Taking \(d_i\) smaller if necessary, we assume that

Given any \(d<d_i\), we take a homotopy \(\xi ^u,\,u \in [0,i]\), of \(K^{i-1}\)-plane fields given by step i and more specifically satisfying for all \((t, u) \in K^{i-1} \times [0,i]\) and all \(p\in U\)

By the homotopy extension property, we can extend this homotopy to \(K^i\). According to (\(\measuredangle _0\)) and (\(\ddagger _i\)), for all st in the same simplex \(K_*\) of \(K^i\) and all \(p \in U\),

Furthermore, according to (\(\dagger _i\)) and (\(\diamond _i\)), for all \(t\in K^i\) and all \((p,q)\in (A_d)^2\) such that \(|p-q|<2d\),

$$\begin{aligned} \angle (\xi ^i_t(p),\xi ^i_t(q))\le \left|| \xi ^i_t \right||_1 |q-p| \le \frac{\theta }{16 d_i} \times 2d < \frac{ \theta }{ 8 }. \end{aligned}$$

So \(\xi ^i\) satisfies the hypothesis of Claim 6.18. Thus, for any \(K_*\), special simplices (for \(\xi ^i\)) are disjoint.

From now on we fix an i-simplex \(K_*\) (not contained in L).

Deformation near special simplices

We first define the coordinates in which we want to apply the deformation model (after rescaling), then Claim 6.20 specifies the setting to which we can actually apply it, and Claim 6.22 makes sure the deformation can be made relative to \(F_d\). These two assertions are fairly straightforward for constant plane fields, and the point is that the more we shrink the scaling factor d, the more the rescaled plane fields are close to being constant. Finally we keep track of the amplitude of the deformation to make sure that, at the end of step \(i+1\), the new plane fields still satisfy the angle conditions necessary for their special simplices to be disjoint.

Let \(\sigma _d\) be a special simplex (for \(\xi ^i\)), \(V_d\) its \(d\delta /2\)-neighborhood and \(\alpha _d\) any edge of \(\sigma _d\) if \(\sigma _d \cap F_d=\varnothing \) and the (single) edge \(\sigma _d \cap F_d\) otherwise (cf. Remark 6.16). We now choose adapted coordinates in the following way: the origin is the midpoint q of \(\alpha _d\), the vector \(\partial _y(q)\) is tangent to \(\sigma _d\) and points to the vertex opposite \(\alpha _d\), and the vector \(\partial _x(q)\) is orthogonal to \(\sigma _d\) (Fig. 19).

Fig. 19
figure19

Choice of coordinates near a special simplex \(\sigma _d\)

Now denote by \(h_d\) the homothety of ratio d and center q, by \(V,\,\sigma ,\,\alpha \) and \(\bar{F}\) the inverse images of \(V_d,\,\sigma _d,\,\alpha _d\) and \(F_d\) under \(h_d\), and by \(\zeta _t^i,\,t \in K_*\), the pull-back \((h_d)^*\xi ^i_t\) defined on V. With the above choices, \(\zeta ^i_t,\,t \in K_*\), satisfies condition (\(*\)) of Section 6.4 with \(\tilde{\theta }= \theta \) since, for all \((t,p) \in K_* \times N_1 (\sigma ) \supset K_* \times V\),

$$\begin{aligned} \angle ({\zeta _t^i}^\perp (p), \partial _x(p))=\angle ({\xi _t^i}^\perp (h_d(p)), \partial _x(h_d(p))) = \angle ( \xi ^i_t(h_d(p)), \sigma ) < \theta \end{aligned}$$

according to Claim 6.18. In addition, according to (\(\dagger _i\)) and (\(\diamond _i\)),

$$\begin{aligned} \left\| \zeta _i^t \right\| _1 = \left\| \xi ^i_t \right\| _{d,1}\le d_i c_{i,1} <\frac{\theta }{16}<1. \end{aligned}$$

As in Sect. 6.4, \(\eta \) denotes the plane field defined by \(dz=0\). By the induction hypothesis, the plane field \(\xi ^i_t\) is integrable on \(N_{\mu _i d}(\sigma _d)\) for \(t \in \partial K_*\) and on \(N_{\mu _i d}(F_d)\) for \(t \in K_*\), so \(\zeta ^i_t\) is integrable on \(N_{\mu _i}(\sigma )\) for \(t \in \partial K_*\) and on \(N_{\mu _i}(\bar{F})\) for \(t \in K_*\). We denote by S the smooth boundary of some stricly convex domain containing \(N_{9 \mu _i /10}(\sigma )\) and contained in \(N_{\mu _i }(\sigma )\).

Claim 6.20

There are positive numbers \(d_{i+1/4},\,\mu \) and \(\kappa \) such that if \(d < d_{i+1/4}\), for all \(t\in K_*\), the surface S contains a disk \(D_t\) varying continuously with t and satisfying the following properties:

  • \(D_t\) is transverse to \(\zeta ^i_t \cap \eta \) and its orbit segments under \(\zeta ^i_t \cap \eta \) cover the \(2\mu \)-neighborhood of \(\sigma \) ;

  • \(D_t \cap P\) is connected for every leaf P of \(\eta \) and its angle with \(\zeta ^i_t \cap \eta \) is at least \(\kappa \).

Remark 6.21

The constants \(d_{i+1/4},\,\mu \) and \(\kappa \) depend only on \(\theta \) and the geometry of S, i.e. on \(\theta ,\,\mu _i\) and the geometry of the model simplices of \(\Delta \). They do not depend on the specific simplex \(\sigma \) itself (Fig. 20).

Fig. 20
figure20

Schematic picture for Claim 6.20

Proof

For every \(t\in K_*\), denote by \(\bar{\zeta }^i_t\) the constant plane field equal to \(\zeta ^i_t(q)\) on V. Then \(\bar{\zeta }^i_t\cap \eta \) is tangent to S along a simple closed curve which divides S into an “entrance face” \(S_t^-\) and an “exit face” \(S_t^+\), which depend continuously on t. The flow lines of \(\bar{\zeta }^i_t\cap \eta \) through \(S_t^-\) cover the whole domain B bounded by S and hence \(N_{9 \mu _i /10}(\sigma )\). So if one defines \(D_t\) as \(S_t^-\) with an \(\varepsilon \)-neighborhood of its boundary removed, for some small fixed \(\varepsilon \) (independent of \(t,\,\sigma \)...), the flow lines through \(D_t\) still cover a \(2\bar{\mu }\)-neighborhood of \(\sigma \) (for some \(\bar{\mu }\) depending only on \(\mu _i\) and \(\varepsilon \)) and for every leaf P of \(\eta ,\,P\cap D_t\) is connected and its angle with \(\bar{\zeta }^i_t\cap \eta \) is bounded below by some constant \(\bar{\kappa }\) depending only on \(\varepsilon \) and some lower bound on the curvatures of S (this lower bound depending only on \(\mu _i\) and the global geometry of \(\Delta \), not on the simplex \(\sigma \) under scrutiny).

Now taking \(\mu \) and \(\kappa \) slightly smaller than \(\bar{\mu }\) and \(\bar{\kappa }\), the two properties of Claim 6.20 are satisfied provided \(\zeta _t^i\) is sufficiently \(C^0\)-close to \(\bar{\zeta }^i_t\) (in terms of the geometric constants), which can be guaranteed by assuming d to be small enough, since \(\left\| \zeta ^i_t \right\| _1\le d c_{i,1}\). \(\square \)

In the following claim, we consider the (hard) case where \(\sigma _d\cap F_d\) is nonempty, and thus \(\alpha _d=\sigma _d\cap F_d\).

Claim 6.22

Taking \(\mu \) and \(d_{i+1/4}\) smaller if necessary, the orbit segments of \(\zeta ^i_t \cap \eta \) starting from \(D_t\) and entirely contained in \(N_{\mu _i}(\alpha )\) cover \(N_{2 \mu }(\sigma ) \cap N_{\mu }(\bar{F})\), for every \(t \in K_*\).

Proof

First observe that given \(\tilde{\mu }\in [0,\mu _i]\), there exists \(\mu \) (depending only on \(\tilde{\mu }\) and the geometry of \(\Delta \)) such that

$$\begin{aligned} N_{2 \mu }(\sigma ) \cap N_{\mu }(\bar{F}) \subset N_{\tilde{\mu }}(\alpha ). \end{aligned}$$

So what we actually need to find is \(\tilde{\mu }\in [0,\mu _i]\) such that \(\zeta ^i_t\) satisfies the following property (provided d has been chosen small enough):

The union of orbit segments of \(\zeta ^i_t \cap \eta \) starting from \(D_t\) and entirely contained in \(N_{\mu _i }(\alpha )\) contains \(N_{\tilde{\mu }}(\alpha )\).

As in the proof of Claim 6.20, denote by \(\bar{\zeta }^i_t\) the constant plane field equal to \(\zeta ^i_t(q)\) on V. The union \(U_t\) of the orbit segments of \(\bar{\zeta }^i_t \cap \eta \) starting from \(D_t\) and entirely contained in \(N_{\mu _i}(\alpha )\) is simply the intersection of the orbits of \(\bar{\zeta }^i_t \cap \eta \) starting from \(D_t\) with \(N_{\mu _i}(\alpha )\), since these orbits are straight lines and \(N_{\mu _i}(\alpha )\) is convex. In particular, according to Claim 6.20, \(U_t\) contains \(N_{2\mu }(\sigma )\cap N_{\mu _i}(\alpha )\), which contains \(N_{\bar{\mu }}(\alpha )\) provided \(\bar{\mu }<\min (2\mu ,\mu _i)\).

Now taking \(\tilde{\mu }\) slightly smaller than \(\bar{\mu },\,\zeta ^i_t\) satisfies the above property in italics provided \(\zeta _t^i\) is sufficiently \(C^0\)-close (in terms of the geometric constants) to \(\bar{\zeta }^i_t\), which again can be guaranteed by assuming that d is small enough. \(\square \)

From now on we assume \(d \le d_{i+1/4}\). Claim 6.20 shows that the hypotheses of Lemma 6.6 are satisfied by every plane field \(\zeta ^i_t,\,t \in K_*\), for the constants \(\mu \) and \(\kappa \). Setting \(\mu _{i+1/4} = \mu \), we thus obtain a homotopy \(\zeta ^u,\,u \in [i,i+1/4]\), of \(K_*\)-plane fields, so that:

  • \(\zeta _t^u\) coincides with \(\zeta _t^i\) outside of \(N_{2 \mu _{i+1/4} }(\sigma )\);

  • \(\zeta _t^{i+1/4}\) is integrable on the \(\mu _{i+1/4}\)-neighborhood of \(\sigma \) ;

  • for every \(m\ge 1,\,\left\| \zeta ^u_t \right\| _m\le \chi _m(\left\| \zeta ^i_t \right\| _{m+1})\) for all \((t,u) \in K_* \times [i,i+1/4]\), for some universal polynomial \(\chi _m\) with positive coefficients and no constant term. Now

    $$\begin{aligned} \left\| \zeta ^i_t \right\| _{m+1} = \left\| \xi ^i_t \right\| _{d,m+1}\le c_{i,m+1}d \end{aligned}$$

    and there exists a constant \(c_{i+1/4,m}\ge c_{i,m}\) such that \(\chi _m(c_{i,m+1}x)\le c_{i+1/4,m}x\) for all \(x\le d_{i+1/4}\) (cf. Remark 6.8), so in the end \(\left\| \zeta ^u_t \right\| _m\le c_{i+1/4,m} d\).

Remark 6.23

The constants \(c_{i+1/4,m}\) depend on \(\theta ,\,\kappa ,\,\mu \) and the geometry of \(\Delta \), but not on \(\sigma \) itself.

Hence the rescaling \(\xi ^u,\,u \in [i,i+1/4]\), of \(\zeta ^u\) by a factor d has the following properties:

  • \(\xi _t^u\) coincides with \(\xi _t^i\) outside of \(N_{2 \mu _{i+1/4} d}(\sigma _d)\);

  • \(\xi _t^{i+1/4}\) is integrable on the \(\mu _{i+1/4} d\)-neighborhood of \(\sigma _d\);

  • for every \(m\ge 1\), there exists a constant \(c_{i+1/4,m}\) such that \(||\xi _t^u ||_{d,m}\le c_{i+1/4,m}d\) for all \((t,u) \in K_* \times [0,i+1/4]\).

Reducing \(d_{i+1/4}\) if necessary so that

we can assume the angle variation of each \(\xi ^u_t\) on \(V_d = N_{d \delta /2}(\sigma )\) is less than \(\beta /8\). Then for all \((t,u) \in K_* \times [i,i+1/4]\) and all \(p \in V_d\),

Indeed, if \(q \in V_d{\setminus }N_{2 \mu _{i+1/4} d}(\sigma )\),

$$\begin{aligned} \angle (\xi ^u_t(p), \xi ^i_t(p))\le & {} \angle (\xi ^u_t(p), \xi ^u_t(q)) \\&+ \angle (\xi ^u_t(q), \xi ^i_t(q)) + \angle (\xi ^i_t(q), \xi ^i_t(p)) < \frac{\beta }{8} + 0 + \frac{\beta }{8}. \end{aligned}$$

Inequalities (\(\measuredangle _0\)) and (\(\diamond \diamond _{i+1/4}\)) imply, for all \((t,u) \in K_* \times [0,i+1/4]\) and all \(p \in U\),

For \(t \in \partial K_*\), Remark 6.15 shows that the homotopy \(\zeta ^u_t\) (and thus \(\xi ^u_t\)), \(u \in [i,i+1/4]\), is completely stationary. Indeed, the intersection of the flow cylinder of \(D_t\) with the domain bounded by S (which contains the support of the homotopy) is an interval fiber bundle over \(D_t\) on which \(\zeta ^i_t\) is assumed to be already integrable for every \(t \in \partial K_* \subset K^{i-1}\).

Moreover, the same remark together with Claim 6.22 shows that for every \(t \in K_*\), the homotopy \(\xi ^u_t\) is stationary on \(N_{\mu _{i+1/4}d}(F_d)\).

Since the neighborhoods \(V_d=N_{d\delta /2}(\sigma )\) of the different special simplices \(\sigma \) are disjoint (by definition of \(\delta \) and according to Claim 6.18), we can apply Lemma 6.6 to all of them simultaneously and we obtain constants \(d_{i+1/4},\,\mu _{i+1/4}\) and \(c_{i+1/4,m}\) independent of \(\sigma \) (cf. Remarks 6.21 and 6.23).

Deformation near the other simplices

0-simplices.  Let q be a vertex of \(A_d\) belonging neither to \(F_d\) nor to any special simplex, and \(V_d\) its \(d \delta /2\)-neighborhood. Note that \(V_d\) is disjoint from the \(d \delta /2\)-neighborhoods of the special simplices, so that \(\xi ^{i+1/4}_t\) coincides with \(\xi ^i_t\) on \(V_d\) for every \(t \in K_*\). Again, denote by \(h_d\) the homothety of factor d and center q, by V the inverse image of \(V_d\), and by \(\zeta _t^{i+1/4},\,t \in K_*\), the pull-back \((h_d)^*\xi ^{i+1/4}_t\) defined on V. For S, take the intersection of V with a plane perpendicular to \(\zeta ^{i+1/4}_s(q)\) for some \(s \in K_*\). Define the coordinate axes as follows:

  • \(\partial _y(q) \in \zeta ^{i+1/4}_s(q)\) is orthogonal to S;

  • \(\partial _x(q) \in T_qS\) is orthogonal to \(\zeta ^{i+1/4}_s(q)\) (Fig. 21).

Fig. 21
figure21

Choice of coordinates near a vertex q

Combining (\(\measuredangle _0\)), (\(\diamond _{i+1/4}\)) and (\(\ddagger _{i+1/4}\)), one can check that condition (\(*\)) (cf. Sect. 6.4) is satisfied by every \(\zeta ^{i+1/4}_t,\,t \in K_*\), for \(\tilde{\theta }= \theta /8\). With these notations, one easily proves an analogue of Claim 6.20, which provides numbers \(d_{i+1/2},\,\mu = \mu _{i+1/2},\,\kappa = \pi /2 - \theta /8\) and disks \(D_t\) which can be taken to be independent of t and contained in the \(\mu _{i+1/4}\)-neighborhood of q. From now on we assume \(d\le d_{i+1/2}\). Lemma 6.6 then gives a homotopy \(\zeta ^u,\,u \in [i+1/4,i+1/2]\), of \(K_*\)-plane fields whose rescaling \(\xi ^u\) by a factor d has the following properties:

  • \(\xi _t^u\) coincides with \(\xi _t^{i+1/4}\) outside of \(N_{2 \mu _{i+1/2} d}(q)\);

  • \(\xi _t^{i+1/2}\) is integrable on the \(\mu _{i+1/2} d\)-neighborhood of q;

  • for every \(m\ge 1\), there is a constant \(c_{i+1/2,m}\) such that \(||\xi _t^u ||_{d,m}\le c_{i+1/2,m}d\) for all \((t,u) \in K_* \times [0,i+1/2]\).

Reducing \(d_{i+1/2}\) if necessary so that

one can make sure (as for special simplices) that for all \((t,u) \in K_* \times [i+1/4,i+1/2]\) and all \(p \in V\),

Inequalities (\(\ddagger _{i+1/4}\)) and (\(\diamond \diamond _{i+1/2}\)) imply that for all \((t,u) \in K_*\times [0,i+1/2]\),

For \(t \in \partial K_*\), Remark 6.15 shows once again that the homotopy \(\zeta ^u_t\) (and thus \(\xi ^u_t\)), \(u \in [i+1/4,i+1/2]\), is completely stationary, since \(D_t\) is contained in \(N_{\mu _{i+1/4}}(q)\). As for special simplices, we apply Lemma 6.6 simultaneously near all vertices.

1-simplices.  We now consider an edge \(\alpha _d\) of \(A_d\) contained neither in \(F_d\) nor in any special simplex and we denote by q its midpoint and by \(V_d\) its \(d \delta /2\)-neighborhood. Again, denote by \(h_d\) the homothety of factor d and center q, by V and \(\alpha \) the inverse images of \(V_d\) and \(\alpha _d\) under \(h_d\), and by \(\zeta _t^{i+1/2},\,t \in K_*\), the pull-back \((h_d)^*\xi ^{i+1/2}_t\) defined on V. For the surface S, we take the intersection of V with a plane that contains \(\alpha \) and is perpendicular to \(\zeta ^{i+1/2}_s(q)\) for some \(s\in K_*\). The coordinate axes are defined as follows: the vector \(\partial _y(q) \in \zeta ^{i+1/2}_s(q)\) is orthogonal to S and the vector \(\partial _x(q) \in T_qS\) is orthogonal to \(\zeta ^{i+1/2}_s(q)\) (Fig. 22).

According to (\(\measuredangle _0\)), (\(\diamond _{i+1/2}\)) and (\(\ddagger _{i+1/2}\)), condition (\(*\)) is satisfied by every \(\zeta ^{i+1/2}_t,\,t \in K_*\), with \(\tilde{\theta }= \theta /8\). Here again, one can prove an analogue of Claim 6.20, which provides numbers \(d_{i+3/4},\,\mu = \mu _{i+3/4},\,\kappa = \pi /2 - \theta /8\) and disks \(D_t\) which can be taken independent of t and contained in the \(\mu _{i+1/2}\)-neighborhood of \(\alpha \). From now on we assume \(d\le d_{i+3/4}\).

Fig. 22
figure22

Choice of coordinates near an edge \(\alpha \)

Lemma 6.6 then gives a homotopy \(\zeta ^u,\,u \in [i+1/2,i+3/4]\), of \(K_*\)-plane fields whose rescaling \(\xi ^u\) by a factor d satisfies:

  • \(\xi _t^u\) coincides with \(\xi _t^{i+1/2}\) outside of \(N_{2 \mu _{i+3/4} d}(\alpha _d)\);

  • \(\xi _t^{i+3/4}\) is integrable on the \(\mu _{i+3/4} d\)-neighborhood of \(\alpha _d\);

  • for every \(m\ge 1\), there is a constant \(c_{i+3/4,m}\) such that \(||\xi _t^u ||_{d,m}\le c_{i+3/4,m}d\) for all \((t,u) \in K_* \times [0,i+3/4]\),

and analogues \(\diamond _{i+3/4},\,\diamond \diamond _{i+3/4}\) and \(\ddagger _{i+3/4}\) of \(\diamond _{i+1/2},\,\diamond \diamond _{i+1/2}\) and \(\ddagger _{i+1/2}\) respectively. In particular, for every \(t \in \partial K_*\), Remark 6.15 shows once again that the homotopy \(\xi ^u_t,\,u \in [i+1/2,i+3/4]\), is completely stationary. Moreover, for every \(t \in K_*\), every integral curve of \(\zeta ^{i+1/2}_t \cap \eta \) intersecting \(D_t\) meets \(N_{\mu _{i+1/2}}(\partial \alpha )\) along an interval, for \(N_{\mu _{i+1/2}}(\partial \alpha )\) is made of two strictly convex balls. It then follows from Remark 6.15 that \(\xi ^u_t = \xi ^{i+1/2}_t\) on \(N_{\mu _{i+1/2} d}(\partial \alpha _d)\) for all \(u \in [i+1/2,i+3/4]\). In other words, the deformation changes nothing in the \(\mu _{i+1/2} d\)-neighborhood of the 0-skeleton. One can thus perform the deformations simultaneously near all edges.

(non-special) 2-simplices.  Finally, let \(\sigma _d\) be a non-special face of \(A_d\) not contained in \(F_d,\,V_d\) its \(d \delta /2\)-neighborhood and q its center. Again, denote by \(h_d\) the homothety of factor d and center q, by V and \(\sigma \) the inverse images of \(V_d\) and \(\sigma _d\) under \(h_d\), and by \(\zeta _t^{i+3/4},\,t \in K_*\), the pull-back \((h_d)^*\xi ^{i+3/4}_t\) defined on V. For the surface S, take the intersection of V with the plane containing \(\sigma \). We fix the coordinate axes as follows:

  • \(\partial _y(q)\) belongs to \(\zeta ^{i+3/4}_s(q)\) for some \(s \in K_*\) and has maximal angle with \(\sigma \) ;

  • \(\partial _x(q)\) is orthogonal to \(\zeta ^{i+3/4}_s(q)\) (Fig. 23).

Here again, every \(\zeta ^{i+3/4}_t,\,t \in K_*\), satisfies condition (\(*\)) for \(\tilde{\theta }= \theta /8\).

Fig. 23
figure23

Choice of coordinates near a face \(\sigma \)

Moreover, since \(\sigma \) is non-special, \(\angle (\zeta ^i_t(p), \sigma ) \ge \theta /2\) for all \((t,p) \in K_* \times \sigma \). This, together with inequalities (\(\diamond \diamond _{i+k/4}\)), \(1 \le k \le 3\), and (\(\diamond _{i+3/4}\)) (left as an exercise to the reader) implies that \(\angle (\zeta _t^{i+3/4}(p), \sigma ) \ge \theta /2 - \beta \ge \theta /4\) for all \((t,p) \in K_* \times V\). This lower bound allows us to prove an analogue of Claim 6.20, which provides constants \(d_{i+1},\,\mu = \mu _{i+1},\,\kappa = \theta /4\) and disks \(D_t\) which can be taken independent of t and contained in the \(\mu _{i+3/4}\)-neighborhood of \(\sigma \). We may assume \(d\le d_{i+1}\). Lemma 6.6 then gives a homotopy \(\zeta ^u,\,u \in [i+3/4,i+1]\), of \(K_*\)-plane fields whose rescaling \(\xi ^u\) by a factor d has properties analogous to the ones we listed for \(u\in [0,3/4]\). Since every integral curve of \(\zeta ^{i+3/4}_t \cap \eta \) which intersects \(D_t\) meets the \(\mu _{i+3/4}\)-neighborhood of each edge of \(\sigma \) along an interval, the deformation does not affect \(N_{\mu _{i+3/4}}(\partial \sigma )\). One can thus once again (and for the last time) make the modifications simultaneously on all faces.

Carrying out this construction for every i-simplex \(K_*\) of \(K^i\), we finally obtain a homotopy \(\xi ^u,\,u \in [0,i+1]\), of \(K^i\)-plane fields on U with all the properties needed to conclude step \(i+1\) of the induction.

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Eynard-Bontemps, H. On the connectedness of the space of codimension one foliations on a closed 3-manifold. Invent. math. 204, 605–670 (2016). https://doi.org/10.1007/s00222-015-0622-8

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Mathematics Subject Classification

  • 57R30