Asymptotic strictly pseudoconvex CR structure for asymptotically locally complex hyperbolic manifolds

Abstract

In this paper, we build a compactification by a strictly pseudoconvex CR structure for a complete and non-compact Kähler manifold whose curvature tensor is asymptotic to that of the complex hyperbolic space. To do so, we study in depth the evolution of various geometric objects that are defined on the leaves of some foliation of the complement of a suitable convex subset, called an essential subset, whose leaves are the equidistant hypersurfaces above this latter subset. With a suitable renormalization which is closely related to the anisotropic nature of the ambient geometry, the above mentioned geometric objects converge near infinity, inducing the claimed structure on the boundary at infinity.

1 Introduction

The study of the asymptotic geometry of complete non-compact Riemannian manifolds have proven to be fruitful in the understanding of the geometry of complex domains, driven by the following remark. By endowing the interior of a bounded domain with a complete metric, which then sends its boundary to infinity, one can read much information on the geometry of the boundary in the asymptotic development of the metric, see [14, 15, 21]. The induced geometric structure on the boundary leads to geometric invariants of the domain. The Bergman metric and the Kähler-Einstein metric are examples of such metrics, and have been at the centre of complex geometry for decades. On the unit ball of ${\mathbb {C}}^n$, these two latter metrics are equal up to a multiplicative constant. This example is particularly interesting, and is called the complex hyperbolic space: it is the unique simply connected and complete Kähler manifold with constant holomorphic sectional curvature equal to $-1$. In that sense, it is the complex counterpart of the real hyperbolic space. In polar coordinates in an exponential chart, the complex hyperbolic metric is given by the following expression

$$\begin{aligned} {{\,\textrm{d}\,}}\!r^2 + 4 \sinh ^2(r)\theta \otimes \theta + 4\sinh \left( \frac{r}{2}\right) \gamma , \end{aligned}$$

where $\theta$ is the standard contact form of the unit sphere of ${\mathbb {C}}^n$, and $\gamma = {{\,\textrm{d}\,}}\!\theta (\cdot ,i\cdot )$ is the Levi form induced on the contact distribution $\ker \theta$. This form of the metric reveals the geometric structure of the odd dimensional sphere, which is a strictly pseudoconvex Cauchy–Riemann (CR) manifold. We say that the CR sphere is the sphere at infinity of the complex hyperbolic sphere. This example is a particular case of the more general non-compact rank one symmetric spaces, namely the real hyperbolic spaces, the complex hyperbolic spaces, the quaternionic hyperbolic spaces and the octonionic hyperbolic plane. They all admit a sphere at infinity, which is endowed with a particular geometric structure closely related to the Riemannian metric of these spaces; they are called conformal infinities. Asymptotically hyperbolic Einstein metrics with prescribed conformal infinity have been built by Graham and Lee [19, 23]. The general case of asymptotically symmetric metrics have been covered by Biquard [6]. Similar results, yet unpublished, have been obtained simultaneously by Roth in his Ph.D thesis for the complex hyperbolic case [25].

In a series of papers [2,3,4, 18], Bahuaud et al. have studied a converse problem in the real hyperbolic setting. They give therein intrinsic geometric conditions under which a complete non-compact Riemannian manifold whose sectional curvature decays sufficiently fast to $-1$ near infinity have a conformal boundary at infinity, modelled on that of the hyperbolic space. This generalizes a work of Anderson and Schoen [1]. These geometric conditions are the existence of a convex core, called essential subset, and the convergence near infinity of the sectional curvature to $-1$, and of some covariant derivatives of the curvature tensor to 0, with exponential decays.

By trying to determine which complete Kähler manifolds arise as bounded submanifolds of some Hermitian space ${\mathbb {C}}^N$, Bland has studied an analogous problem in the Kähler case, and has built a compactification by a CR structure for some complete non-compact manifolds whose curvature tensor is asymptotic to that of the complex hyperbolic space in some particular coordinates [9, 10]. Nonetheless, his assumptions appear to be rather restrictive, and not completely geometric: roughly, it is asked that the curvature tensor, with a suitable renormalization, extends up to the boundary in some already compactified coordinates, with high regularity. They imply, in particular, the a posteriori estimates $R = R^0 + {\mathcal {O}}(e^{-3r})$ and $\nabla R = {\mathcal {O}}(e^{-4r})$, where $R^0$ is the constant $-1$ holomorphic sectional curvature tensor, and r is the distance function to some compact subset. However, Biquard and Herzlich have shown in [7] that, in real dimension 4, the curvature tensor R of an asymptotically complex hyperbolic Einstein metric has the following asymptotic development

$$\begin{aligned} R = R^0 + C e^{-2r} + {\mathcal {O}}(e^{-2r}), \end{aligned}$$

where C turns out to be a (non-zero) multiple of the Cartan tensor of the CR structure at infinity. Since the Cartan tensor of a CR structure vanishes exactly when the CR structure is spherical (that is, if it is locally CR diffeomorphic to the unit sphere), it seems that Bland’s results only hold for a few examples of asymptotically complex hyperbolic metrics, at least in real dimension 4 and in the Einstein setting.

More recently, it has been shown by Bracci et al. that a convex domain in ${\mathbb {C}}^n$ with boundary of class at least ${\mathcal {C}}^{2,\alpha }$ ($\alpha >0$) is strictly pseudoconvex if and only if one can endow this domain with a complete Kähler metric whose holomorphic sectional curvature has range in $[-1-\varepsilon ,-1+\varepsilon ]$ near the boundary [11, 28]. Here, the constant $\varepsilon$ only depends on the dimension and the regularity of the boundary. In this case, the holomorphic sectional curvature has the form $-1 + {\mathcal {O}}(e^{-ar})$ for some $a>0$, where r is the distance function from a compact subset. Let us mention that this result has been proven to be false if the regularity of the boundary is only ${\mathcal {C}}^{2}$ [16].

Inspired by the work of Bahuaud et al., we give in this paper a geometric characterization of complete non-compact Kähler manifolds admitting a compactification by a strictly pseudoconvex CR structure. The study is more intricate in the complex hyperbolic setting than in the real hyperbolic one, due to the anisotropic nature of complex hyperbolic geometry. In contrast with Bland’s results, our study only relies on purely geometric assumptions, and requires a control of the curvature and its first covariant derivative to an order strictly less than ${\mathcal {O}}(e^{-2r})$. We consider a complete non-compact Kähler manifold (M, g, J) whose Riemann curvature tensor R is asymptotic to $R^0$, the curvature tensor of constant holomorphic sectional curvature $-1$. We first prove that if R is exponentially close to $R^0$ near infinity, then the metric tensor has an asymptotic development at infinity that is similar to that of the complex hyperbolic space.

Theorem A

Let (M, g, J) be a complete non-compact Kähler manifold with an essential subset K. Assume that the sectional curvature of $\overline{M\setminus K}$ is negative and that there exists $a>1$ such that $\Vert R-R^0\Vert _g = {\mathcal {O}}(e^{-ar})$. Then $\partial \!K$ is endowed with a nowhere vanishing continuous differential 1-form $\eta$ and a continuous field of symmetric positive semi-definite bilinear forms $\gamma _H$, positive definite on the distribution $H=\ker \eta$, such that the metric reads in an exponential chart

$$\begin{aligned} {{\,\textrm{d}\,}}\!r^2 + e^{2r} \eta \otimes \eta + e^r \gamma _H + \text {lower order terms}. \end{aligned}$$

See Sect. 4.7 for a precise statement. In view of the computations, and due to the characterization of strictly pseudoconvex domains by Bracci et al., the decay rate of the curvature is a reasonable assumption. Such a Kähler manifold will be referred to as an asymptotically locally complex hyperbolic manifold. The local condition on the curvature tensor is thus a sufficient condition to recover a development of the metric similar to that of the model space. The differential form $\eta$ is called the canonical 1-form at infinity, and the field of symmetric bilinear forms $\gamma _H$ is called the Carnot–Carathéodory metric at infinity. Under the extra condition that the metric is asymptotically locally symmetric, we can moreover prove the following.

Theorem B

Let (M, g, J) be a complete non-compact Kähler manifold satisfying the assumptions of Theorem A. Assume furthermore that there exists $b >1$ such that $\Vert \nabla R\Vert _g = {\mathcal {O}}(e^{-br})$. Then the canonical 1-form at infinity $\eta$ is a contact form of class ${\mathcal {C}}^1$.

Under higher exponential decays, the Carnot–Carathéodory metric also gains one order of regularity.

Theorem C

Let (M, g, J) be a complete non-compact Kähler manifold satisfying the assumptions of Theorem B. Assume furthermore that $\min \{a,b\}>\frac{3}{2}$. Then the Carnot–Carathéodory metric at infinity $\gamma _H$ is of class ${\mathcal {C}}^1$.

We then show that under the assumptions of Theorem C, the contact distribution H is endowed with a canonical integrable almost complex structure $J_H$ having the same properties as that of the model space.

Theorem D

Let (M, g, J) be a complete non-compact Kähler manifold satisfying the assumptions of Theorem C. Then the contact distribution at infinity $H=\ker \eta$ is endowed with an integrable almost complex structure $J_H$ at infinity of class ${\mathcal {C}}^1$ such that the Carnot–Carathéodory metric at infinity is given by $\gamma _H = {{\,\textrm{d}\,}}\!\eta (\cdot ,J_H\cdot )$. In particular, $(\partial \!K, H,J_H)$ is a strictly pseudoconvex CR manifold of class ${\mathcal {C}}^1$.

1.1 Structure of the paper

In Sect. 2, we detail the notations and the setting. In Sect. 3, we define the asymptotically locally complex hyperbolic (ALCH) and asymptotically locally symmetric (ALS) conditions and give a lower bound on the volume growth. In Sect. 4, we study the asymptotic behaviour of normal Jacobi fields, and then prove Theorem A. Section 5 is devoted to the proof of Theorems B and C. Section 6 is finally dedicated to the definition of the almost complex structure at infinity and to the proof of Theorem D. Useful curvature computations can be found in the Appendix A.

2 Preliminaries

2.1 Notations

Let $(M^{2n+2},g,J)$ be a complete, non-compact Kähler manifold of real dimension $2n+2$, $n\geqslant 1$. We denote by $\nabla$ its Levi-Civita connection. We recall that (M, g, J) is Kähler if the almost complex structure J is parallel and satisfies $g(JX,JY) = g(X,Y)$ for all tangent vectors X and Y. The Riemann curvature tensor R is defined by

$$\begin{aligned} R(X,Y)Z = \nabla _{[X,Y]}Z - \big (\nabla _X(\nabla _YZ)-\nabla _Y(\nabla _XZ)\big ). \end{aligned}$$

(2.1.1)

By abuse of notation, its four times covariant version is still denoted by R, that is $R(X,Y,Z,T) = g(R(X,Y)Z,T)$. Please note that our convention is that of [5, 17], which is opposite to that of [13, 24]. In our case, the sectional curvature of a linear plane P with orthonormal basis $\{X,Y\}$ is $\sec (P) = \sec (X,Y)=R(X,Y,X,Y)$ and the holomorphic sectional curvature is given by R(X, JX, X, JX).

Let $K\subset M$ be a compact codimension 0 submanifold with hypersurface boundary $\partial \!K$ oriented by a unit normal $\nu$. The associated outward normal exponential map $E:{\mathbb {R}}_+ \times \partial \!K\rightarrow M$ is defined by $E(r,p) = \gamma _p(r)$, where $\gamma _p$ is the unit speed geodesic with initial data $\gamma _p(0) = 0$ and $\gamma _p'(0) = \nu (p)$. Following [4], the submanifold K is called an essential subset if $\partial \!K$ is convex with respect to $\nu$, meaning that its shape operator is non-negative, and if $E:{\mathbb {R}}_+ \times \partial \!K\rightarrow \overline{M{\setminus } K}$ is a diffeomorphism. If K is totally convex and if the sectional curvature outside of K is negative, then K is an essential subset (see [4, Theorem 3.1]). Moreover, the visual boundary of (M, g) is homeomorphic to the boundary of K.

Assume that $K\subset M$ is an essential subset. The radial vector field $\partial _r$ is the vector field on $\overline{M\setminus K}$ defined by $\partial _r(\gamma _p(r)) = E(r,p)_*\frac{{{\,\textrm{d}\,}}\!}{{{\,\textrm{d}\,}}\!r}= \gamma _p'(r)$. A tensor defined on $\overline{M\setminus K}$ is said to be radially parallel if its covariant derivative in the $\partial _r$ direction vanishes. Since $(\gamma _p)_{p\in \partial \!K}$ are geodesics, $\partial _r$ is radially parallel. The shape operator S and the Jacobi operator $R_{\partial _r}$ are the fields of symmetric endomorphisms defined by $SX = \nabla _X\partial _r$ and $R_{\partial _r}X = R(\partial _r,X)\partial _r$. They are related by the Riccati equation

$$\begin{aligned} {\mathcal {L}}_{\partial _r}S = \nabla _{\partial _r}S = - S^2 - R_{\partial _r}. \end{aligned}$$

(2.1.2)

They both vanish on $\partial _r$ and take values in $\{\partial _r\}^{\perp }$. From the Riccati equation we derive the following: if $\partial \!K$ is convex and the sectional curvature of $\overline{M\setminus K}$ is non-positive, then the level hypersurfaces above $\partial \!K$ are convex.

If $v \in T_p\partial \!K$ is a tangent vector, the associated normal Jacobi field along $\gamma _p$ is defined as $Y_v(r) = E(r,p)_*v$. If v is a (local) vector field on $\partial \!K$, we write $Y_v(\gamma _p(r))= Y_{v(p)}(r)$, which is still called a normal Jacobi field.

Lemma 2.1

If v is a (local) vector field on $\partial \!K$, then:

• 1. The vector fields $Y_v$ and $\partial _r$ commute.

• 2. $\nabla _{\partial _r}Y_v = SY_v$.

• 3. $Y_v$ is everywhere orthogonal to $\partial _r$.

• 4. The restriction of $Y_v$ along any radial geodesic $\gamma _p$ is a Jacobi field.

Proof

• 1. Since E is a diffeomorphism onto its image, the naturality of the Lie bracket yields $[\partial _r,Y_v] = E_*[\frac{{{\,\textrm{d}\,}}\!}{{{\,\textrm{d}\,}}\!r},v] = E_* 0 = 0$.

• 2. The Levi-Civita connection is torsion-free, hence $SY_v - \nabla _{\partial _r}Y_v =[\partial _r,Y_v]$. The result follows by the first point.

• 3. The equation of geodesics $\nabla _{\partial _r}\partial _r= 0$ together with the second point give $\partial _r(g(Y_v,\partial _r)) = g(SY_v,\partial _r)$. This last expression identically vanishes since S has range in $\{\partial _r\}^{\perp }$. Hence, the function $g(Y_v,\partial _r)$ is constant along radial geodesics, and the result follows from the initial condition $g(v,\nu ) = 0$.

• 4. From the second point, it holds that $\nabla _{\partial _r}Y_v = SY_v$. One thus obtains the equality $\nabla _{\partial _r}(\nabla _{\partial _r}Y_v) =\nabla _{\partial _r}(SY_v) = (\nabla _{\partial _r}S)Y_v + S \nabla _{\partial _r}Y_v$, and the result follows from the Riccati equation.

$\square$

Recall that (M, g, J) is assumed to be Kähler. We define the constant $-1$ holomorphic sectional curvature tensor $R^0$ by

$$\begin{aligned} \begin{aligned} R^0(X,Y,Z,T)&= \frac{1}{4}\big ( g(X,T)g(Y,Z) - g(X,Z)g(Y,T)\\&\quad + g(X,JT)g(Y,JZ) - g(X,JZ)g(Y,JT)\\&\quad + 2g(X,JY)g(T,JZ)\big ). \end{aligned} \end{aligned}$$

It is a parallel tensor having the symmetries of a curvature tensor. In case (M, g, J) is the complex hyperbolic space, then $R=R^0$. Note that this definition differs from that of [22, IX.7] by its sign: the reason is that we require for $R^0$ to have $-1$ holomorphic sectional curvature. For any orthonormal pair of vectors fields $\{X,Y\}$, $R^0(X,Y,X,Y) = -\frac{1}{4}(1 + 3g(JX,Y)^2)$, from which is deduced the fundamental pinching

$$\begin{aligned} -1 \leqslant R^0(X,Y,X,Y) \leqslant - \frac{1}{4}. \end{aligned}$$

If $P = {{\,\textrm{span}\,}}\{X,Y\}$, the lower bound is achieved exactly when $P = JP$ (we say that P is a complex line) while the upper bound is achieved exactly when $P\perp JP$ (we say that P is a totally real plane).

2.2 Contact and CR manifolds

A contact form on an $(2n+1)$-dimensional manifold M is a differential form of degree one $\alpha$ such that $\alpha \wedge {{\,\textrm{d}\,}}\!\alpha ^n \ne 0$ everywhere. The associated contact structure is $H=\ker \alpha$. We say that (M, H) is a contact manifold, and that $\alpha$ calibrates H. Note that for H fixed, $\alpha$ is not unique. The Reeb vector field of $\alpha$ is the unique vector field $X_{\alpha }$ such that $\alpha (X_{\alpha }) = 1$ and ${{\,\textrm{d}\,}}\!\alpha (X_{\alpha },\cdot )=0$.

An almost complex structure J on (M, H) is a section of the bundle ${{\,\textrm{End}\,}}(H)$ such that $J^2 = -{{\,\textrm{Id}\,}}_H$. The complexified bundle $H\otimes {\mathbb {C}}$ splits as $H \otimes {\mathbb {C}}= H^{1,0}\oplus H^{0,1}$ into the eigenspaces of the complex linear extension of J, where $H^{1,0} = \{X-iJX \mid X \in H\}$ and $H^{0,1} = \{X+iJX \mid X \in H\}$. The almost complex structure J is said to be integrable if the sections of $H^{1,0}$ form a linear subalgebra of the sections of $TM\otimes {\mathbb {C}}$, that is if $[X,Y] \in \Gamma (H^{1,0})$ whenever $X,Y\in \Gamma (H^{1,0})$.

A CR manifold (M, H, J) is a contact manifold (M, H) endowed with an integrable almost complex structure on H. When $H = \ker \alpha$ and ${{\,\textrm{d}\,}}\!\alpha (\cdot ,J\cdot )$ is positive definite on H, $(M,\alpha ,J)$ is called a strictly pseudoconvex CR manifold. CR geometry is the natural geometry of real hypersurfaces in complex manifolds. The toy model is ${\mathbb {S}}$, the unit sphere of ${\mathbb {C}}^n$, with contact distribution $H = T{\mathbb {S}}\cap (iT{\mathbb {S}})$ and with almost complex structure J given by the multiplication by i in the fibres. Endowed with its natural contact form, it is a strictly pseudoconvex CR manifold.

2.3 Some analysis results

Throughout the paper, we will use the following two theorems from real analysis several times.

Theorem

(Grönwall’s inequality) Let $I=[t_0,T) \subset {\mathbb {R}}$ be an interval with $T\in (t_0,+\infty ]$ and let $\varphi ,\alpha$ and $\beta :I \rightarrow {\mathbb {R}}$ be respectively continuous, non-decreasing, and non-negative continuous functions. Assume that

$$\begin{aligned} \forall t \in I,\quad \varphi (t) \leqslant \alpha (t) + \int _{t_0}^t \beta (s)\varphi (s) {{\,\textrm{d}\,}}\!s. \end{aligned}$$

(2.3.1)

Then it holds that

$$\begin{aligned} \forall t \in I,\quad \varphi (t) \leqslant \alpha (t) \exp \left( \int _{t_0}^t \beta (s){{\,\textrm{d}\,}}\!s\right) . \end{aligned}$$

An inequality of the form (2.3.1) will be referred to as a “Grönwall-like inequality".

Theorem

(Rademacher’s Theorem) Let $\varphi :I \rightarrow {\mathbb {R}}$ be a locally Lipschitz function defined on an interval I, and let $t_0\in I$ be fixed. Then $\varphi$ is almost everywhere differentiable, and it holds that

$$\begin{aligned} \forall t \in I,\quad \varphi (t) = \varphi (t_0) + \int _{t_0}^t \varphi '(s){{\,\textrm{d}\,}}\!s, \end{aligned}$$

where the latter integral is the Lebesgue integral of the almost everywhere defined function $\varphi '$.

Rademacher’s Theorem will be applied to the restriction of the norm of tensors restricted along radial geodesics $\gamma _p$.

The following Lemma will be useful in order to estimate the growth of normal Jacobi fields.

Lemma 2.2

Let $f :{\mathbb {R}}_+ \rightarrow {\mathbb {R}}$ be a ${\mathcal {C}}^2$ function. Assume that there exist three positive constants $\alpha , \beta$ and $\gamma$ such that

$$\begin{aligned} \forall t \geqslant 0,\quad |f''(t) + \alpha f'(t)| \leqslant \gamma e^{-\beta t}. \end{aligned}$$

Then f has a limit $f_{\infty }\in {\mathbb {R}}$ when $t \rightarrow +\infty$ and it holds that

$$\begin{aligned} \begin{aligned} \forall t \geqslant 0,\quad |f'(t)|&\leqslant {\left\{ \begin{array}{ll} \left( |f'(0)| + \frac{\gamma }{\beta -\alpha }\right) e^{-\beta t} &{} \text {if } \beta< \alpha , \\ \left( |f'(0)| + \gamma t \right) e^{-\alpha t} &{} \text {if } \beta = \alpha , \\ \left( |f'(0)| + \frac{\gamma }{\beta -\alpha } \right) e^{-\alpha t} &{} \text {if } \beta> \alpha , \end{array}\right. }\\ \forall t \geqslant 0,\quad |f(t)-f_{\infty }|&\leqslant {\left\{ \begin{array}{ll} \left( |f'(0)| + \frac{\gamma }{\beta -\alpha }\right) \frac{e^{-\beta t}}{\beta } &{} \text {if } \beta < \alpha , \\ \frac{\alpha |f'(0)| + \gamma (\alpha t+1)}{\alpha ^2} e^{-\alpha t} &{} \text {if } \beta = \alpha , \\ \left( |f'(0)| + \frac{\gamma }{\beta -\alpha } \right) \frac{e^{-\alpha t}}{\alpha } &{} \text {if } \beta > \alpha . \end{array}\right. } \end{aligned} \end{aligned}$$

Proof

Write $e^{\alpha t}(f''(t)+\alpha f'(t)) = \left( e^{\alpha t} f'(t)\right) '$. Integrating on [0, t] and using the assumption yields

$$\begin{aligned} \forall t \geqslant 0, \quad |f'(t)| \leqslant e^{-\alpha t} |f'(0)| + \gamma e^{-\alpha t}\int _0^t e^{(\alpha -\beta )x} {{\,\textrm{d}\,}}\!x. \end{aligned}$$

(2.3.2)

There are three cases to consider depending on $\beta < \alpha$, $\beta =\alpha$ or $\beta > \alpha$. In any case, equation (2.3.2) shows that $f'$ is integrable and hence that f converges. The result follows from a straightforward integration. $\square$

We finally give a Lemma that will be applied to give a uniform bound on the norm of the shape operator.

Lemma 2.3

Let $\sigma :{\mathbb {R}}_+ \rightarrow {\mathbb {R}}$ be a locally Lipschitz function. Let C and a be positive constants such that the following inequality holds almost everywhere

$$\begin{aligned} \sigma ' \leqslant -\sigma ^2 + 1 + C e^{-at}. \end{aligned}$$

Then there exists a constant $C'>0$ depending only on C and a such that

$$\begin{aligned} \forall t \geqslant 0,\quad \sigma (t) \leqslant 1 + (\sigma (0)+C') {\left\{ \begin{array}{ll} e^{-at} &{} \text {if } 0<a<2,\\ (t+1)e^{-2t} &{} \text {if } a =2,\\ e^{-2t} &{} \text {if } a >2. \end{array}\right. } \end{aligned}$$

Proof

Let $\varsigma = \sigma - 1$. Then $\varsigma$ is locally Lipschitz and almost everywhere differentiable, and it follows from the assumption on $\sigma$ and from the fact that $\varsigma ^2 \geqslant 0$, that

$$\begin{aligned} \varsigma '(t) \leqslant -2 \varsigma + Ce^{-at} \quad a.e. \end{aligned}$$

Thus, it holds that

$$\begin{aligned} (\varsigma (t)e^{2t})' \leqslant Ce^{(2-a)t} \quad a.e. \end{aligned}$$

Integrating this last inequality yields

$$\begin{aligned} \forall t \geqslant 0,\quad \varsigma (t) \leqslant \varsigma (0)e^{-2t} + C{\left\{ \begin{array}{ll} \frac{e^{-at}-e^{-2t}}{2-a} &{} \text {if } 0<a<2,\\ t e^{-2t} &{} \text {if } a = 2,\\ \frac{e^{-2t}-e^{-at}}{a-2}&{} \text {if } a >2. \end{array}\right. } \end{aligned}$$

The result now follows from comparing the exponents. $\square$

3 Asymptotically locally complex hyperbolic manifolds

3.1 The ALCH and ALS conditions

We define the two following asymptotic conditions.

Definition 3.1

((ALCH) and (ALS) manifolds) Let (M, g, J) be a complete non-compact Kähler manifold, $K\subset M$ be a compact subset and $r = d_g(\cdot ,K)$ be the distance function to K.

• 1. (M, g, J) is said to be asymptotically locally complex hyperbolic, (ALCH) in short, of order $a>0$, if there exists a constant $C_0>0$ such that

  $$\begin{aligned} \Vert R-R^0\Vert _g \leqslant C_0 e^{-ar}. \end{aligned}$$

• 2. (M, g) is said to be asymptotically locally symmetric, (ALS) in short, of order $b>0$, if there exists a constant $C_1 >0$ such that

  $$\begin{aligned} \Vert \nabla R\Vert _g \leqslant C_1 e^{-br}. \end{aligned}$$

Remark 3.2

These two definitions do not depend on the choice of the compact subset $K \subset M$.

In practice, K will refer to an essential subset. The complex hyperbolic space is of course (ALCH) and (ALS) of any order since in that case, $R=R^0$ and $\nabla R=0$.

3.2 First consequences

We fix (M, g, J) an (ALCH) manifold of order $a>0$. The following Lemmas are direct consequences of the definition of $R^0$.

Lemma 3.3

The norm of the Riemann curvature tensor of an (ALCH) manifold of order $a>0$ is uniformly bounded.

Lemma 3.4

Let $p\in M$ and $P\subset T_pM$ be a tangent plane. Then

$$\begin{aligned} -1 - C_0 e^{-ar(p)} \leqslant \sec (P) \leqslant -\frac{1}{4} + C_0e^{-ar(p)}. \end{aligned}$$

We now assume that $K\subset M$ is an essential subset and that the sectional curvature of $\overline{M\setminus K}$ is non-positive. In that case, the shape operator S is positive semi-definite on $\overline{M\setminus K}$, and its operator norm at a point $\gamma _p(r)\in \overline{M\setminus K}$ is given by its largest eigenvalue.

Proposition 3.5

There exists a constant $C>0$ independent of (r, p) such that

$$\begin{aligned} \Vert S_{\gamma _p(r)}\Vert _g \leqslant 1 + C {\left\{ \begin{array}{ll} e^{-ar} &{} \text {if } 0<a<2,\\ (r+1)e^{-2r} &{} \text {if } a =2,\\ e^{-2r} &{} \text {if } a >2. \end{array}\right. } \end{aligned}$$

In particular, $\Vert S\Vert _g$ is uniformly bounded on $\overline{M{\setminus } K}$, and $\Vert S\Vert _g-1$ is bounded above by an integrable function on $\overline{M\setminus K}$.

Proof

Let $p \in \partial \!K$ and $\sigma :{\mathbb {R}}\rightarrow {\mathbb {R}}$ be defined by $\sigma (r) = \Vert S_{\gamma _p(r)}\Vert _g$. Since S is positive semi-definite, $\sigma (r)$ is the largest eigenvalue of $S_{\gamma _p(r)}$. Identify all tangent spaces along $\gamma _p$ using parallel transport, and let $(S_r)_{r\geqslant 0}$ and $(R_r)_{r\geqslant 0}$ be the associated endomorphisms of $T_p\partial \!K$ obtained from S and $R_{\partial _r}$. Therefore, $\sigma$ is given by the expression $\sigma (r) = \sup _{X\in T_p\partial \!K, \Vert X\Vert _g=1} g(S_rX,X)$. In this identification, the Riccati equation reads $S_r' = -S_r^2 - R_r$. Let $X\in T_p\partial \!K$ be a unit vector, $r\geqslant 0$ and $h>0$. Then

$$\begin{aligned} \begin{aligned} g(S_{r+h}X,X)&= g\big ((S_r + h S_r' + o(h))X,X\big ) \\&= g\big ((S_r-hS_r^2)X,X\big ) + h (-g(R_rX,X) + o(1)\big ). \end{aligned} \end{aligned}$$

For small enough $h>0$, $S_r - hS_r^2$ is positive semi-definite and its largest eigenvalue is given by $\sigma (r) - h\sigma (r)^2$. In addition, $g(R_rX,X)$ is the sectional curvature of the tangent plane spanned by $\partial _r$ and the parallel transport of X along $\gamma _p$ evaluated at $\gamma _p(r)$, which is bounded below by $-1-C_0e^{-ar}$ (Lemma 3.4). It then follows that

$$\begin{aligned} g(S_{r+h}X,X) \leqslant \sigma (r) - h\sigma (r)^2 + h\left( 1+C_0e^{-ar} + o(1)\right) . \end{aligned}$$

Taking the supremum over all unit vectors X of this last inequality yields

$$\begin{aligned} \sigma (r+h) \leqslant \sigma (r) - h\sigma (r)^2 + h\left( 1+C_0e^{-ar} + o(1)\right) , \end{aligned}$$

from which is deduced the inequality

$$\begin{aligned} \limsup _{h\rightarrow 0} \frac{\sigma (r+h) - \sigma (r)}{h} \leqslant - \sigma (r)^2 + 1 + C_0e^{-ar}. \end{aligned}$$

(3.2.1)

Since S is smooth and $\Vert \cdot \Vert$ is Lipschitz, $\sigma$ is locally Lipschitz, and by Rademacher’s Theorem, $\sigma$ is almost everywhere differentiable. It then follows from Eq. (3.2.1) that

$$\begin{aligned} \sigma ' \leqslant -\sigma ^2 + 1 + C_0e^{-ar} \quad a.e. \end{aligned}$$

According to Lemma 2.3, there exists $C'>0$ depending only on $C_0$ and a such that

$$\begin{aligned} \Vert S_{\gamma _p(r)}\Vert _g \leqslant 1 + (\Vert S_p\Vert _g + C'){\left\{ \begin{array}{ll} e^{-ar} &{} \text {if } 0<a<2,\\ (r+1)e^{-2r} &{} \text {if } a =2,\\ e^{-2r} &{} \text {if } a >2. \end{array}\right. } \end{aligned}$$

The result now follows by defining $C = C' + \sup _{p\in \partial \!K} \Vert S_p\Vert _g$ which is finite by compactness of $\partial \!K$. $\square$

3.3 A volume growth lower bound

The (ALCH) assumption implies two bounds on the sectional curvature. The lower bound $\sec \geqslant -1 + {\mathcal {O}}(e^{-ar})$ forces the largest eigenvalue of the shape operator, and thus its operator norm, to be uniformly bounded. We shall now show that the upper bound $\sec \leqslant -\frac{1}{4} + {\mathcal {O}}(e^{-ar})$ forces the trace of the shape operator to be bounded from below. We then derive a lower bound on the volume density function using Eq. (3.3.5). Our study is inspired by [12, 20].

We first show the following general Lemma from geometric comparison analysis. It is very similar to different results stated in [20]. The author could not find a precise proof of the exact formulation given below and therefore decided to give one, relying on classical Jacobi field techniques.

Lemma 3.6

Let $(M^{m+1},g)$ be a complete Riemannian manifold, $N\subset M$ be a hypersurface co-oriented by a unit normal $\nu$, $p\in \partial \!K$, and $\kappa >0$ such that

• 1. N is convex with respect to $\nu$,

• 2. there exists $R>0$ such that the normal exponential map E is a diffeomorphism from $[0,R)\times N$ onto its image,

• 3. the sectional curvature on this image is non-positive,

• 4. there exists $r_0 \in [0,R)$ such for all $r\in [r_0,R)$, the sectional curvature of any linear plane tangent at $\gamma _p(r)$ is bounded above by $-\kappa ^2$.

Then it holds that

$$\begin{aligned} \forall r \in [r_0,R),\quad {{\,\textrm{trace}\,}}(S_{\gamma _p(r)}) \geqslant m\kappa \tanh \big (\kappa (r-r_0)\big ). \end{aligned}$$

Proof

The proof is quite long and goes in two steps. We first bound from below the trace of S by an integral related to the norm of some Jacobi fields. Then, we compare the situation with its counterpart in the hyperbolic space of dimension $m+1$ and sectional curvature $-\kappa ^2$, where N is replaced by an isometrically embedded m dimensional hyperbolic space of the same sectional curvature.

Let $\{\partial _r,E_1,\ldots ,E_m\}$ be an orthonormal frame along $\gamma _p$ obtained using the parallel transport of an orthonormal basis $\{\nu ,e_1,\ldots ,e_m\}$ of $T_pM$. Let $r \in [r_0,R)$ be fixed. For $j \in \{1,\ldots ,m\}$, let $Y_j$ be the unique Jacobi field along $\gamma _p$ such that $Y_j(r) = E_j(r)$ and $\nabla _{\partial _r}Y_j(r) = S_{\gamma _p(r)}E_j(r)$. It is shown similarly to the proof of Lemma 2.1 that the equality $\nabla _{\partial _r}Y_j = SY_j$ holds all along $\gamma _p$. By assumptions 2. and 3. and by the convexity of N, S is a positive definite operator and it follows that, for $j\in \{1,\ldots ,m\}$

$$\begin{aligned} g(SY_j(r),Y_j(r)) \geqslant g(SY_j(r),Y_j(r)) - g(SY_j(r_0),Y_j(r_0)) = \int _{r_0}^{r}\frac{{{\,\textrm{d}\,}}\!}{{{\,\textrm{d}\,}}\!t} g(SY_j,Y_j). \end{aligned}$$

The Jacobi field equation yields the equality

$$\begin{aligned} \frac{{{\,\textrm{d}\,}}\!}{{{\,\textrm{d}\,}}\!t}g(SY_j,Y_j) = \Vert \nabla _{\partial _r}Y_j\Vert _g^2 - \sec (\partial _r,Y_j)\Vert Y_j\Vert _g^2. \end{aligned}$$

Since ${{\,\textrm{trace}\,}}\big (S_{\gamma _p(r)}\big ) = \sum _{j=1}^m g(SE_j(r),E_j(r))$, the initial conditions for the Jacobi fields $Y_j$ and the assumption 4. now gives

$$\begin{aligned} {{\,\textrm{trace}\,}}\big (S_{\gamma _p(r)}\big ) \geqslant \sum _{j=1}^m \int _{r_0}^{r} \Vert \nabla _{\partial _r}Y_j\Vert _g^2 +\kappa ^2\Vert Y_j\Vert _g^2. \end{aligned}$$

(3.3.1)

We now compare the right-hand side of (3.3.1) with a similar situation in the space form of curvature $-\kappa ^2$. Let $({\overline{M}},{\overline{g}})= {\mathbb {R}}H^{m+1}(-\kappa ^2)$ be that space form, ${\overline{\nabla }}$ its Levi-Civita connection, ${\overline{N}}= {\mathbb {R}}H^m(-\kappa ^2) \hookrightarrow {\overline{M}}$ be isometrically embedded, for instance as the equatorial hyperplane in the ball model, with a unit normal ${\overline{\nu }}$, and shape operator ${\overline{S}}$. Let $\{{\overline{\partial _r}},{\overline{E}}_1,\ldots ,{\overline{E}}_m\}$ be an orthonormal frame along a radial geodesic ${{\overline{\gamma }}}_{{\overline{p}}}$ obtained using the parallel transport of an orthonormal basis $\{{\overline{\nu }},{\overline{e}}_1,\ldots ,{\overline{e}}_m\}$ of $T_{{\overline{p}}}{\overline{M}}$. For $j\in \{1,\ldots ,m\}$, let ${\overline{Y}}_{\! j}$ be the unique Jacobi field along ${{\overline{\gamma }}}_{{\overline{p}}}$ such that ${\overline{\nabla }}_{{\overline{\partial _r}}}{\overline{Y}}_{\! j} = {\overline{S}}{\overline{Y}}_{\! j}$ and ${\overline{Y}}_{\! j}(r-r_0) = {\overline{E}}_j(r-r_0)$. Finally, let $X_j$ be defined by

$$\begin{aligned} \forall t \in [0,r-r_0],\quad X_j(t) = \sum _{k=1}^m g\left( Y_j(r_0+t),E_k(r_0+t)\right) {\overline{E}}_k(t), \end{aligned}$$

so that the following equalities hold

$$\begin{aligned} \begin{aligned} \forall t \in [0,r-r_0],\quad \Vert X_j(t)\Vert _{{\overline{g}}}&= \Vert Y_j(r_0+t)\Vert _g,\\ \Vert {\overline{\nabla }}_{{\overline{\partial _r}}}X_j(t)\Vert _{{\overline{g}}}&= \Vert \nabla _{\partial _r}Y_j(r_0+t)\Vert _g. \end{aligned} \end{aligned}$$

(3.3.2)

Expanding the inequality $\Vert {\overline{\nabla }}_{{\overline{\partial _r}}}({\overline{Y}}_{\! j}-X_j)\Vert _{{\overline{g}}}^2 + \kappa ^2 \Vert {\overline{Y}}_{\! j} - X_j\Vert _{{\overline{g}}} \geqslant 0$, and integrating the developed expression on $[0,r-r_0]$ yields

$$\begin{aligned} \begin{aligned} \int _0^{r-r_0} \Vert {\overline{\nabla }}_{{\overline{\partial _r}}}X_j\Vert _{{\overline{g}}}^2 + \kappa ^2\Vert X_j\Vert _{{\overline{g}}}^2&\geqslant 2 \int _0^{r-r_0}{\overline{g}}({\overline{\nabla }}_{{\overline{\partial _r}}}{\overline{Y}}_{\! j},{\overline{\nabla }}_{{\overline{\partial _r}}}X_j) + \kappa ^2 {\overline{g}}({\overline{Y}}_{\! j},{X}_j) \\ {}&\quad - \int _0^{r-r_0}\Vert {\overline{\nabla }}_{{\overline{\partial _r}}}{\overline{Y}}_{\! j}\Vert _{{\overline{g}}}^2 + \kappa ^2 \Vert {\overline{Y}}_{\! j}\Vert _{{\overline{g}}}^2. \end{aligned} \end{aligned}$$

(3.3.3)

Since ${\overline{Y}}_{\! j}$ is a Jacobi field, the integrands of the right-hand side satisfy

$$\begin{aligned} \begin{aligned} {\overline{g}}({\overline{\nabla }}_{{\overline{\partial _r}}}{\overline{Y}}_{\! j},{\overline{\nabla }}_{{\overline{\partial _r}}}X_j) + \kappa ^2 {\overline{g}}({\overline{Y}}_{\! j},{X}_j)&= \frac{{{\,\text {d}\,}}\!}{{{\,\text {d}\,}}\!t} {\overline{g}}({\overline{\nabla }}_{{{\overline{\partial _r}}}}{\overline{Y}}_{\! j},X_j), \text{ and }\\ \Vert {\overline{\nabla }}_{{\overline{\partial _r}}}{\overline{Y}}_{\! j}\Vert _{{\overline{g}}}^2 + \kappa ^2 \Vert {\overline{Y}}_{\! j}\Vert _{{\overline{g}}} ^2&= \frac{{{\,\text {d}\,}}\!}{{{\,\text {d}\,}}\!t}{\overline{g}}({\overline{\nabla }}_{{\overline{\partial _r}}}{\overline{Y}}_{\!\! j},{\overline{Y}}_{\!\! j}). \end{aligned} \end{aligned}$$

Recall that ${\overline{\nabla }}_{{\overline{\partial _r}}}{\overline{Y}}_{\! j} = {\overline{S}}{\overline{Y}}_{\! j}$. Since N is isometrically embedded, it holds that ${\overline{S}}_{{\overline{p}}}=0$. Also, recall that at $r-r_0$, ${\overline{Y}}_{\! j}(r-r_0) = X_j(r-r_0)$. Hence, (3.3.3) yields

$$\begin{aligned} \int _0^{r-r_0} \Vert {\overline{\nabla }}_{{\overline{\partial _r}}}X_j\Vert _{{\overline{g}}}^2 + \kappa ^2\Vert X_j\Vert _{{\overline{g}}}^2 \geqslant {\overline{g}}({\overline{S}}{\overline{E}}_j(r-r_0),{\overline{E}}_j(r-r_0)). \end{aligned}$$

(3.3.4)

It now follows from equations (3.3.1), (3.3.2) and (3.3.4) that we have the inequality ${{\,\textrm{trace}\,}}\big (S_{\gamma _p(r)}\big ) \geqslant {{\,\textrm{trace}\,}}\big ( {\overline{S}}_{{\overline{\gamma }}_{{\overline{p}}}(r-r_0)}\big )$. Note that $\Sigma _{{{\overline{\gamma }}}_{{\overline{p}}}(t)} = \kappa \tanh (\kappa t) {{\,\textrm{Id}\,}}_{{\{{{\overline{\gamma }}}_{{\overline{p}}}(t)\}}^{\perp }}$ satisfies the Riccati equation

$$\begin{aligned} {\overline{\nabla }}_{{\overline{\partial _r}}} \Sigma =- \Sigma ^2 - {\overline{R}}_{{\overline{\partial _r}}}, \end{aligned}$$

with initial data $\Sigma _{{\overline{p}}}=0$, so that ${\overline{S}}=\Sigma$. This concludes the proof. $\square$

We now return to the setting of this paper and consider a Kähler manifold (M, g, J) with an essential subset K. It is canonically oriented, and so is the co-oriented hypersurface $\partial \!K$. Let $v_g$ and $v_{g|_{\partial \!K}}$ be the Riemannian volume forms induced by the metrics g and $g|_{\partial \!K}$. Since E is a diffeomorphism, $E^*v_g$ and ${{\,\textrm{d}\,}}\!r \wedge v_{g|_{\partial \!K}}$ are two volume forms on ${\mathbb {R}}_+\times \partial \!K$, and they are proportional. The volume density function $\lambda$ is the unique positive function $\lambda :{\mathbb {R}}_+\times \partial \!K\rightarrow {\mathbb {R}}$ such that $E^*v_g = \lambda {{\,\textrm{d}\,}}\!r \wedge v_{g|_{\partial \!K}}$. It is clear that $\lambda |_{\{0\}\times \partial \!K}=1$. The volume density function and the shape operator are related by the following differential equation

$$\begin{aligned} \frac{\partial \lambda (r,p) }{\partial r} = \lambda (r,p) \,{{\,\textrm{trace}\,}}(S_{\gamma _p(r)}). \end{aligned}$$

(3.3.5)

We shall now derive from Lemma 3.6 a lower bound on the volume density function.

Proposition 3.7

Let $(M^{2n+2},g,J)$ be an (ALCH) manifold of order $a>0$ with an essential subset K, such that the sectional curvature of $\overline{M\setminus K}$ is negative. Let $\varepsilon \in (0,n + \frac{1}{2})$ be fixed. Then there exist $r_0=r_0(\varepsilon ) >0$ and $\Lambda _-= \Lambda _-(\varepsilon ) >0$ such that the volume density function $\lambda$ satisfies

$$\begin{aligned} \forall (r,p) \in [r_0,+\infty )\times \partial \!K,\quad \lambda (r,p) \geqslant \Lambda _- e^{(n+\frac{1}{2}-\varepsilon )r}. \end{aligned}$$

Proof

Let $\kappa = \frac{1}{2}- \frac{\varepsilon }{2n+1}>0$ and $r_0 = \max \left\{ \frac{1}{a}\ln \big (\frac{\frac{1}{4}-\kappa ^2}{C_0}\big ),1\right\} >0$. By definition of $r_0$,

$$\begin{aligned} \forall r \geqslant r_0,\quad -\frac{1}{4} +C_0 e^{-ar} \leqslant -\kappa ^2 <0. \end{aligned}$$

According to Lemma 3.4, the sectional curvature of linear planes based at points $\gamma _p(r)$, with $r\geqslant r_0$, have sectional curvature bounded above by $-\kappa ^2$. Hence, Lemma 3.6 yields the inequality

$$\begin{aligned} \forall (r,p) \in [r_0,+\infty ) \times \partial \!K,\quad {{\,\textrm{trace}\,}}\big ( S_{\gamma _p(r)}\big ) \geqslant (2n+1)\kappa \tanh (\kappa (r-r_0)). \end{aligned}$$

The differential equation (3.3.5) satisfied by $\lambda$ and S now yields

$$\begin{aligned} \forall (r,p) \in [r_0,+\infty ) \times \partial \!K,\quad \frac{\partial _r\lambda (r,p)}{\lambda (r,p)} \geqslant (2n+1)\kappa \tanh (\kappa (r-r_0)). \end{aligned}$$

Integrating this last inequality yields

$$\begin{aligned} \forall (r,p) \in [r_0,+\infty )\times \partial \!K,\quad \lambda (r,p) \geqslant \lambda (r_0,p) \cosh ^{2n+1}(\kappa (r-r_0)). \end{aligned}$$

The result now follows by setting $\Lambda _- = \frac{e^{-(2n+1)\kappa r_0}}{2^{2n+1}} \min _{p\in \partial \!K} \lambda (r_0,p)$, which exists and is positive since $\partial \!K$ is compact and $\lambda$ positive and continuous, and by noticing that $(2n+1)\kappa = n+\frac{1}{2}-\varepsilon$. $\square$

4 Normal Jacobi fields estimates

In this section, we consider $(M^{2n+2},g,J)$ a fixed (ALCH) manifold of order $a>0$ with an essential subset K, and we study the asymptotic behaviour of its normal Jacobi fields. The geometric structure we wish to highlight being of contact nature, we choose not to work in coordinates. Instead, we define some natural moving frames, which we call radially parallel orthonormal frames, in which the computations are convenient.

4.1 Radially parallel orthonormal frame

To the radial vector field $\partial _r$ is naturally associated the vector field $J\!\partial _r$. Since the metric is Kähler, $J\!\partial _r$ is radially parallel. It can be obtained using the parallel transport of $J\nu$ along the radial geodesics $(\gamma _p)_{p\in \partial \!K}$.

Definition 4.1

(Radially parallel orthonormal frame) Let $\{J\nu ,e_1,\ldots ,e_{2n}\}$ be an orthonormal frame defined on an open subset $U\subset \partial \!K$. For $j\in \{1,\ldots ,2n\}$, let $E_j$ be vector fields obtained by the parallel transport of $e_j$ along radial geodesics. The orthonormal frame $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ on the cylinder $E({\mathbb {R}}_+\times U)$ is called the radially parallel orthonormal frame associated to $\{J\nu ,e_1,\ldots ,e_{2n}\}$.

A radially parallel orthonormal frame is composed of radially parallel vector fields. It is worth noting that $\{\partial _r,J\!\partial _r\}$ spans a complex line while $\{\partial _r,E_j\}$ spans a totally real plane if $j\in \{1,\ldots ,2n\}$. Moreover, $\{E_1,\ldots ,E_{2n}\}$ is an orthonormal frame of the J-invariant subbundle $\{\partial _r,J\!\partial _r\}^{\perp }$ of TM.

Lemma 4.2

Let $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame. Then the following holds

$$\begin{aligned} {\left\{ \begin{array}{ll} |\sec (\partial _r,J\!\partial _r) + 1| \leqslant C_0e^{-ar}, \\ |R(\partial _r,J\!\partial _r,\partial _r,E_j)| \leqslant C_0e^{-ar}, &{} \forall j\in \{1,\ldots ,2n\},\\ |\sec (\partial _r,E_j) + \frac{1}{4}| \leqslant C_0e^{-ar}, &{} \forall j\in \{1,\ldots ,2n\},\\ |R(\partial _r,E_i,\partial _r,E_j)| \leqslant C_0e^{-ar}, &{} \forall i,j\in \{1,\ldots ,2n\}, i\ne j. \end{array}\right. } \end{aligned}$$

Proof

Let X and Y be unit vector fields. The (ALCH) condition gives

$$\begin{aligned} |R(\partial _r,X,\partial _r,Y) - R^0(\partial _r,X,\partial _r,Y)| \leqslant C_0e^{-ar}. \end{aligned}$$

The result follows from a direct computation of $R^0(\partial _r,X,\partial _r,Y)$ for the different values of X and Y in $\{J\!\partial _r, E_j\}$ and their orthogonal properties. $\square$

4.2 The Jacobi system

Since K is an essential subset, for $r\geqslant 0$, $E(r,\cdot )$ is a diffeomorphism between $\partial \!K$ and the level hypersurface at distance r above $\partial \!K$.

Definition 4.3

(Local 1-forms) Let $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame on the cylinder $E({\mathbb {R}}_+\times U)$. We define on U the family of 1-forms $\{\eta _r,\eta ^1_r\ldots ,\eta ^{2n}_r\}$ by

$$\begin{aligned} \begin{aligned} \forall r \geqslant 0, \quad \eta _r&= e^{-r}E(r,\cdot )^* \left( g(\cdot , J\!\partial _r)|_{{\partial _r}^{\perp }}\right) ,\\ \forall j \in \{1,\ldots ,2n\}, \forall r \geqslant 0,\quad \eta ^j_r&= e^{-\frac{r}{2}} E(r,\cdot )^* \left( g(\cdot , E_j)|_{{\partial _r}^{\perp }}\right) . \end{aligned} \end{aligned}$$

In other words, $\eta _r(v) = e^{-r}g(Y_v,J\!\partial _r)$ and $\eta _r^j(v) = e^{-\frac{r}{2}}g(Y_v,E_j)$. Note that $\eta _r$ is defined on all of $\partial \!K$ and does not depend on the choice of a radially parallel orthonormal frame. With these notations, a normal Jacobi field $Y_v = E_*v$ along a radial geodesic $\gamma _p$ reads

$$\begin{aligned} Y_v = \eta _r(v) e^r J\!\partial _r+ \sum _{j=1}^{2n} \eta _r^j(v) e^{\frac{r}{2}} E_j. \end{aligned}$$

(4.2.1)

On the cylinder $E({\mathbb {R}}_+\times U)$, we also define the following $(2n+1)^2$ functions $\{u^i_k\}_{i,k\in \{0,\ldots ,2n\}}$ by

$$\begin{aligned} u^i_k = - {\left\{ \begin{array}{ll} \sec (\partial _r,J\!\partial _r)+1 &{} \text {if } i=k=0,\\ e^{-\frac{r}{2}} R(\partial _r,J\!\partial _r,\partial _r,E_k) &{} \text {if } i = 0,k\in \{1,\ldots ,2n\},\\ e^{\frac{r}{2}}R(\partial _r,E_i,\partial _r,J\!\partial _r) &{} \text {if } i\in \{1,\ldots ,2n\}, k = 0,\\ R(\partial _r,E_i,\partial _r,E_k) &{} \text {if } i,k\in \{1,\ldots ,2n\}, i\ne k,\\ \sec (\partial _r,E_i) + \frac{1}{4} &{} \text {if } i=k\in \{1,\ldots , 2n\}. \end{array}\right. } \end{aligned}$$

(4.2.2)

For the rest of this subsection, we shall fix the vector v and henceforth drop out the variable v: we will write $\eta _r$ and $\eta ^j_r$ instead of $\eta _r(v)$ and $\eta ^j_r(v)$. In addition, when a summation is involved, we set $\eta ^0_r = \eta _r$ for the sake of compactness.

Lemma 4.4

The functions $\{\eta _r^i(v)\}_{i\in \{0,\ldots ,2n\}}$ are solutions of the linear second order differential system

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _r^2\eta _r + 2 \partial _r\eta _r = \displaystyle \sum _{k=0}^{2n} u^0_k \eta ^k_r,\\ \partial _r^2\eta ^j_r + \partial _r\eta ^j_r = \displaystyle \sum _{k=0}^{2n} u^j_k \eta ^k_r,&{} \forall j \in \{1,\ldots ,2n\}. \end{array}\right. } \end{aligned}$$

(4.2.3)

Proof

Since the vectors of the radially parallel orthonormal frame are radially parallel, it follows that

$$\begin{aligned} \nabla _{\partial _r}(\nabla _{\partial _r}Y_v) = (\partial _r^2\eta _r + 2 \partial _r\eta _r + \eta _r)e^r J\!\partial _r+\sum _{j=1}^{2n}\left( \partial _r^2{\eta ^j_r} + \partial _r\eta ^j_r + \frac{1}{4}\eta ^j_r\right) e^{\frac{r}{2}}E_j. \end{aligned}$$

Since $Y_v$ is a Jacobi field along $\gamma _p$, it holds that

$$\begin{aligned} \nabla _{\partial _r}(\nabla _{\partial _r}Y_v) = -\eta _r e^r R(\partial _r,J\!\partial _r)\partial _r- \sum _{j=1}^{2n} \eta ^j_re^{\frac{r}{2}} R(\partial _r,E_j)\partial _r. \end{aligned}$$

The result then follows from an identification of the coefficients in the orthonormal frame $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$. $\square$

We now give a reformulation of the Jacobi system (4.2.3) with integrals.

Lemma 4.5

The functions $\{\eta ^i_r\}_{i\in \{0,\ldots ,2n\}}$ are solutions of the integral system

$$\begin{aligned} {\left\{ \begin{array}{ll} \eta _r &{}= \quad \displaystyle \eta _0 + \partial _r\eta _0\frac{1-e^{-2r}}{2} +\displaystyle \int _0^r\sum _{k=0}^{2n} u^0_k(\gamma _p(s)) \eta ^k_ss \frac{1-e^{-2(r-s)}}{2}{{\,\textrm{d}\,}}\!s,\\ \eta ^j_r &{}=\quad \eta ^j_0 + \partial _r\eta ^j_0(1-e^{-r}) + \displaystyle \int _0^r\sum _{k=0}^{2n} u^j_k(\gamma _p(s)) \eta ^k_s(1-e^{-(r-s)}){{\,\textrm{d}\,}}\!s, \\ &{} \forall j \in \{1,\ldots ,2n\}. \end{array}\right. } \end{aligned}$$

(4.2.4)

Proof

We give details for the first equation, the other ones being proven similarly. The differential equation satisfied by $\eta _r(v)$ (see equation (4.2.3)) reads

$$\begin{aligned} \partial _r\left( e^{2r}\partial _r\eta _r\right) = \sum _{k=0}^{2n} e^{2r}u^0_k \eta ^k_r, \end{aligned}$$

which integrates as

$$\begin{aligned} \eta _r = \partial _r\eta _0 e^{-2r} + \int _0^r \sum _{k=0}^{2n} u^0_k(\gamma _p(s))\eta ^k_s e^{-2(r-s)} {{\,\textrm{d}\,}}\!s. \end{aligned}$$

A second integration now gives

$$\begin{aligned} \eta _r = \eta _0 + \partial _r\eta _0\frac{1-e^{-2r}}{2} + \sum _{k=0}^{2n}\int _0^r \int _0^tu^0_k(\gamma _p(s))\eta ^k_s e^{-2(t-s)}{{\,\textrm{d}\,}}\!s {{\,\textrm{d}\,}}\!t. \end{aligned}$$

The function $(s,t) \in [0,r]^2 \mapsto u^0_k(\gamma _p(s))\eta ^k_s e^{-2(t-s)}{\textbf{1}}_{\{s\leqslant t\}}$ is measurable and bounded on a compact domain, and hence integrable. It now follows from Fubini’s Theorem that for $k\in \{0,\ldots ,2n\}$, it holds that

$$\begin{aligned} \begin{aligned} \int _0^r \int _0^t u^0_k(\gamma _p(s))\eta ^k_s e^{-2(t-s)} {{\,\textrm{d}\,}}\!s {{\,\textrm{d}\,}}\!t&= \int _{[0,r]^2} u^0_k(\gamma _p(s))\eta ^k_s e^{-2(t-s)}{\textbf{1}}_{\{s\leqslant t\}} {{\,\textrm{d}\,}}\!s {{\,\textrm{d}\,}}\!t \\&= \int _{[0,r]^2}u^0_k(\gamma _p(s))\eta ^k_s e^{-2(t-s)}{\textbf{1}}_{\{s\leqslant t\}} {{\,\textrm{d}\,}}\!t {{\,\textrm{d}\,}}\!s \\&= \int _0^ru^0_k(\gamma _p(s)) \eta ^k_s e^{2s}\left( \int _s^r e^{-2t} {{\,\textrm{d}\,}}\!t\right) {{\,\textrm{d}\,}}\!s \\&= \int _0^r u^0_k(\gamma _p(s)) \eta ^k_s \frac{1-e^{-2(r-s)}}{2}{{\,\textrm{d}\,}}\!s. \end{aligned} \end{aligned}$$

The proof is now complete. $\square$

Let $u :{\mathbb {R}}_+ \rightarrow {\mathbb {R}}_+$ be defined by $u(r) = \max _{i,k\in \{0,\ldots ,2n\}}\left| u^i_k\left( \gamma _p(r)\right) \right|$.

Lemma 4.6

The following upper bound holds

$$\begin{aligned} \forall r\geqslant 0,\quad \sum _{i=0}^{2n} |\eta ^i_r| \leqslant \big ((2n+1)\Vert S_p\Vert +n+1\big )\Vert v\Vert \exp \left( (2n+1)\int _0^r u(s){{\,\textrm{d}\,}}\!s\right) . \end{aligned}$$

Proof

First, note that for $0\leqslant r \leqslant s$, it holds that

$$\begin{aligned} \max \left\{ \left| \frac{1-e^{-2r}}{2}\right| , |1-e^{-r}|, \left| \frac{1-e^{-2(r-s)}}{2}\right| , \left| 1-e^{-(r-s)}\right| \right\} \leqslant 1. \end{aligned}$$

The triangle inequality applied to the integral system (4.2.4) thus yields

$$\begin{aligned} {\left\{ \begin{array}{ll} |\eta _r| &{}\leqslant |\eta _0| + |\partial _r\eta _0| + \displaystyle \int _0^r u(s) \sum _{i=0}^{2n}|\eta ^i_s| {{\,\textrm{d}\,}}\!s,\\ |\eta ^j_r| &{}\leqslant |\eta ^j_0| + |\partial _r\eta ^j_0| +\displaystyle \int _0^r u(s) \sum _{i=0}^{2n}|\eta ^i_s| {{\,\textrm{d}\,}}\!s,\quad \forall j\in \{1,\ldots ,2n\}. \end{array}\right. } \end{aligned}$$

Summing all these inequalities now yields the Grönwall-like inequality

$$\begin{aligned} \sum _{i=0}^{2n} |\eta ^i_r| \leqslant \left( \sum _{i=0}^{2n}|\eta ^i_0| + |\partial _r\eta ^i_0|\right) + \int _0^r (2n+1)u(s) \sum _{i=0}^{2n}|\eta ^i_s| {{\,\textrm{d}\,}}\!s. \end{aligned}$$

Applying Grönwall’s inequality to $\sum _{i=0}^{2n}|\eta ^i_r|$ thus shows that

$$\begin{aligned} \sum _{i=0}^{2n} |\eta ^i_r| \leqslant \left( \sum _{i=0}^{2n} |\eta ^i_0| + |\partial _r\eta ^i_0|\right) \exp \left( (2n+1)\int _0^r u(s) {{\,\textrm{d}\,}}\!s\right) . \end{aligned}$$

Recall that $J\!\partial _r|_{\partial \!K} = J\nu$, and that $E_k(0) = e_k$ for $k\in \{1,\ldots ,2n\}$. Hence, $\eta _0 = g(v,J\nu )$, $\partial _r\eta _0 = g(S_pv-v,J\nu )$, $\eta ^k_0 = g(v,e_k)$, and $\partial _r\eta ^k_0 = g(Sv-\frac{1}{2}v,e_k)$ for $k\in \{1,\ldots ,2n\}$, and the result directly follows from Cauchy-Schwarz inequality. $\square$

We shall now show that if M is (ALCH) of order $a>\frac{1}{2}$, then the components $\{\eta _r^j(v)\}_{j\in \{0,\ldots ,2n\}}$ converge as $r \rightarrow +\infty$ with a well understood decay.

Proposition 4.7

Let $(M^{2n+2},g,J)$ be an (ALCH) manifold of order $a>\frac{1}{2}$ with an essential subset K. Let $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame on a cylinder $E({\mathbb {R}}_+\times U)$, $p\in U$ and $v\in T_p\partial \!K$. Let $Y_v = E_*v$ be the normal Jacobi field along $\gamma _p$ associated to v. If $\eta _r$ and $\{\eta _r^j\}_{j\in \{1,\ldots ,2n\}}$ are the component functions of $Y_v$ defined by equation (4.2.1), then there exists a constant $C>0$ depending only on $C_0$ and a, and constants $\eta _{\infty }$ and $\{\eta ^j_{\infty }\}_{j\in \{1,\ldots ,2n\}}$ such that

$$\begin{aligned} \begin{aligned} \max \{|\partial _r\eta _r|,|\eta _r-\eta _{\infty }|\}&\leqslant C\Vert v\Vert _g {\left\{ \begin{array}{ll} e^{-ar} &{} \text {if } \frac{1}{2}<a<\frac{3}{2},\\ (r+1)e^{-\frac{3}{2}r} &{} \text {if } a= \frac{3}{2},\\ e^{-\frac{3}{2}r} &{} \text {if } a> \frac{3}{2}, \end{array}\right. } \\ \max \{|\partial _r\eta ^j_r|,|\eta ^j_r-\eta ^j_{\infty }|\}&\leqslant C\Vert v\Vert _g {\left\{ \begin{array}{ll} e^{-(a-\frac{1}{2})r} &{} \text {if } \frac{1}{2}<a<\frac{3}{2},\\ (r+1)e^{-r} &{} \text {if } a= \frac{3}{2},\\ e^{-r} &{} \text {if } a > \frac{3}{2}, \end{array}\right. }\\ {}&\hspace{3.9cm} \forall j \in \{1,\ldots ,2n\}. \end{aligned} \end{aligned}$$

(4.2.5)

Proof

According to Lemma 4.2, the definition of the functions $\{u^i_k\}_{i,k\in \{0,\ldots ,2n\}}$ (see Eq. (4.2.2)) yields $0\leqslant u(r) \leqslant C_0 e^{-(a-\frac{1}{2})r}$ for all $r\geqslant 0$. Therefore, u is integrable on ${\mathbb {R}}_+$ and one has $\int _{{\mathbb {R}}_+} u \leqslant \frac{2C_0}{2a-1}$. Since $\partial \!K$ is compact, $\sup _{p\in \partial \!K} \Vert S_p\Vert _g <+\infty$. It follows from Lemma 4.6 that

$$\begin{aligned} \forall r \geqslant 0,\quad \sum _{i=0}^{2n}|\eta ^i_r| \leqslant c\Vert v\Vert _g, \end{aligned}$$

(4.2.6)

with $c= \big ((2n+1)\sup _{p\in \partial \!K}\Vert S_p\Vert _g + n +1\big ) \exp \big (\frac{(4n+2)C_0}{2a-1}\big )$, which only depends on a and $C_0$. Putting this upper bound in the Jacobi system (4.2.3) yields

$$\begin{aligned} {\left\{ \begin{array}{ll} |\partial _r^2\eta _r + 2 \partial _r\eta _r| \leqslant \displaystyle c\Vert v\Vert _g\sum _{k=0}^{2n} |u^0_k|,\\ |\partial _r^2\eta ^j_r + \partial _r\eta ^j_r)| \leqslant \displaystyle c\Vert v\Vert _g\sum _{k=0}^{2n} |u^j_k|, &{}\forall j \in \{1,\ldots ,2n\}. \end{array}\right. } \end{aligned}$$

Looking back at the definitions of the functions $\{u^i_k\}_{i,k\in \{0,\ldots ,2n\}}$ (see Eq. (4.2.2)), Lemma 4.2 gives

$$\begin{aligned} {\left\{ \begin{array}{ll} |\partial _r^2\eta _r + 2 \partial _r\eta _r| &{}\leqslant \displaystyle c'\Vert v\Vert _ge^{-ar},\\ |\partial _r^2\eta ^j_r + \partial _r\eta ^j_r| &{}\leqslant \displaystyle c'\Vert v\Vert _ge^{-(a-\frac{1}{2})r}, \quad \forall j \in \{1,\ldots ,2n\}, \end{array}\right. } \end{aligned}$$

(4.2.7)

with $c' = (2n+1)C_0c$, which only depends on $C_0$ and a. The result now follows from applying Lemma 2.2 to each inequality of the decoupled system (4.2.7), and recalling that we have upper bounds on the initial data $|\eta ^i_0|$ and $|\partial _r\eta ^i_0|$ that are linear in $\Vert v\Vert _g$. Notice that a uniform constant $C>0$ in (4.2.5) can be obtained by taking the maximum of the $(2n+1)$ constants given by Lemma 2.2, which depends only on a and $C_0$. $\square$

Proposition 4.7 yields a pointwise convergence of the coefficients of normal Jacobi fields with a sharp estimate on their asymptotic behaviour. The following subsection focuses on their dependence with respect to the vector v.

4.3 The local 1-forms

We now consider again the dependence in v of the coefficients $\eta _r(v)$ and $\{\eta ^j_r(v)\}_{j\in \{1,\ldots ,2n\}}$. The limits given by Proposition 4.7 are simply denoted by $\eta (v)$ and $\eta ^j(v)$. In these terms, Proposition 4.7 becomes the following.

Proposition 4.8

Let $(M^{2n+2},g,J)$ be an (ALCH) manifold of order $a>\frac{1}{2}$, with an essential subset K. Then there exists a continuous 1-form $\eta$ on $\partial \!K$ and a constant $C>0$ depending only on $C_0$ and a such that $\forall r \in {\mathbb {R}}_+, \forall (p,v) \in T\partial \!K$

$$\begin{aligned} \max \left\{ \left| \partial _r\eta _r(v)\right| ,|\eta _r(v)- \eta (v)|\right\} \leqslant C\Vert v\Vert _g {\left\{ \begin{array}{ll} e^{-ar} &{} \text {if } \frac{1}{2}<a<\frac{3}{2},\\ (r+1)e^{-\frac{3}{2}r} &{} \text {if } a= \frac{3}{2},\\ e^{-\frac{3}{2}r} &{} \text {if } a > \frac{3}{2}. \end{array}\right. } \end{aligned}$$

(4.3.1)

Moreover, if $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ is a radially parallel orthonormal frame on the cylinder $E({\mathbb {R}}_+\times U)$, for all $j \in \{1,\ldots , 2n\}$, there exists a continuous 1-form $\eta ^j$ on U such that $\forall r \in {\mathbb {R}}_+, \forall (p,v) \in TU$

$$\begin{aligned} \max \left\{ \left| \partial _r\eta _r^j(v)\right| ,|\eta ^j_r(v) - \eta ^j(v)|\right\} \leqslant C\Vert v\Vert _g {\left\{ \begin{array}{ll} e^{-(a-\frac{1}{2})r} &{} \text {if } \frac{1}{2}<a<\frac{3}{2},\\ (r+1)e^{-r} &{} \text {if } a= \frac{3}{2},\\ e^{-r} &{} \text {if } a > \frac{3}{2}. \end{array}\right. } \end{aligned}$$

(4.3.2)

Proof

Let $\eta (v) = \eta _{\infty }$ and $\eta ^j(v) = \eta ^j_{\infty }$ for $j\in \{1,\ldots ,2n\}$ be defined pointwise, where $\eta _{\infty }$ and $\{\eta ^j_{\infty }\}_{j\in \{1,\ldots ,2n\}}$ are given by Proposition 4.7. The claimed estimates (4.3.1) and (4.3.2) are obtained as a straightforward translation of Proposition 4.7.

In addition, (4.3.1) shows that $(\eta _r)_{r\geqslant 0}$ uniformly converges to $\eta$ on any compact subset of $T\partial \!K$ as $r\rightarrow +\infty$. Therefore, $\eta$ is continuous. It is furthermore linear in the fibres as a pointwise limit of 1-forms. It follows that $\eta$ is a continuous 1-form on $\partial \!K$. The exact same proof shows that if $j\in \{1,\ldots ,2n\}$, then $(\eta ^j_r)_{r\geqslant 0}$ uniformly converges to $\eta ^j$ on every compact subset of TU as $r\rightarrow +\infty$. Therefore, $\eta ^j$ is a continuous 1-form on U. The proof is now complete. $\square$

Corollary 4.9

Under the assumptions of Proposition 4.8, there exists $c>0$ depending only on a and $C_0$ such that

$$\begin{aligned} \forall (p,v) \in T\partial \!K,\quad |\eta (v)| \leqslant c\Vert v\Vert _g, \end{aligned}$$

and

$$\begin{aligned} \forall j \in \{1,\ldots ,2n\}, \forall (p,v) \in TU,\quad |\eta ^j(v)| \leqslant c \Vert v\Vert _g. \end{aligned}$$

Proof

Equation (4.2.6) gives the existence of c depending only on a and $C_0$ such that

$$\begin{aligned} \forall r \geqslant 0, \forall (p,v) \in TU,\quad |\eta _r(v)| + \sum _{j=1}^{2n} |\eta ^j_r(v)| \leqslant c\Vert v\Vert _g. \end{aligned}$$

The result follows by taking the limit as $r\rightarrow +\infty$. Note that c does not depend on p or U, so that the upper bound is true on all of $\partial \!K$ for $\eta$. $\square$

4.4 Normal Jacobi estimates

Regarding Proposition 4.8, we henceforth assume $a>\frac{1}{2}$. Consider a radially parallel orthonormal frame $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ on a cylinder $E({\mathbb {R}}_+\times U)$, and let $\eta ,\eta ^1,\ldots ,\eta ^{2n}$ be the continuous 1-forms given by Proposition 4.8.

Definition 4.10

(Asymptotic vector fields) For $(r,p)\in {\mathbb {R}}_+\times U$, $v \in T_p\partial \!K$, define the vectors $Z_v(\gamma _p(r)),Z_v'(\gamma _p(r))\in T_{\gamma _p(r)}M$ by

$$\begin{aligned} \begin{aligned} Z_v(\gamma _p(r))&= \eta (v)e^r J\!\partial _r+ \sum _{j=1}^{2n} \eta ^j(v)e^{\frac{r}{2}}E_j,\\ Z_v'(\gamma _p(r))&= \eta (v)e^r J\!\partial _r+ \frac{1}{2}\sum _{j=1}^{2n} \eta ^j(v)e^{\frac{r}{2}}E_j. \end{aligned} \end{aligned}$$

If v is a vector field on U, we refer to $Z_v$ and $Z'_v$ as the asymptotic vector fields related to the vector fields $Y_v$ and $SY_v$.

Proposition 4.11

Assume that $a > \frac{1}{2}$. Let $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame on a cylinder $E({\mathbb {R}}_+\times U)$. Then there exists $C>0$ depending only on a and $C_0$ such that for any vector field v on U,

$$\begin{aligned} \max \{\Vert Y_v-Z_v\Vert _g,\Vert SY_v-Z'_v\Vert _g\} \leqslant C\Vert v\Vert _g {\left\{ \begin{array}{ll} e^{-(a-1)r} &{} \text {if } \frac{1}{2}<a<\frac{3}{2},\\ (r+1)e^{-\frac{r}{2}} &{} \text {if } a = \frac{3}{2},\\ e^{-\frac{r}{2}} &{}\text {if } a>\frac{3}{2}. \end{array}\right. } \end{aligned}$$

Proof

By their very definition, it holds that

$$\begin{aligned} Y_v - Z_v = (\eta _r(v)-\eta (v))e^r J\!\partial _r+ \sum _{j=1}^{2n}(\eta _r^j(v)-\eta ^j(v))e^{\frac{r}{2}}E_j. \end{aligned}$$

Since $SY_v = \nabla _{\partial _r}Y_v$, it also holds that

$$\begin{aligned} SY_v - Z'_v =(\partial _r\eta _r(v)+\eta _r(v)-\eta (v))e^rJ\!\partial _r+ \sum _{j=1}^{2n}\big (\partial _r\eta ^j_r(v)+\frac{1}{2}(\eta ^j_r(v)-\eta ^j(v))\big ) e^{\frac{r}{2}}E_j. \end{aligned}$$

The result follows from the triangle inequality and from Proposition 4.8. $\square$

We conclude this section by stating an important upper bound on the growth of normal Jacobi fields.

Lemma 4.12

There exists a constant $c>0$ depending only on a and $C_0$ such that for any local vector fields v on $\partial \!K$, it holds that

$$\begin{aligned} \forall r\geqslant 0, \forall p,\quad \max \left\{ \Vert Y_v(\gamma _p(r))\Vert _g,\Vert SY_v(\gamma _p(r))\Vert _g\right\} \leqslant c \Vert v\Vert _ge^r. \end{aligned}$$

If moreover, $\eta |_p(v) = 0$, then

$$\begin{aligned} \forall r\geqslant 0,\quad \max \{\Vert Y_v(\gamma _p(r))\Vert _g,\Vert SY_v(\gamma _p(r))\Vert _g\} \leqslant c \Vert v\Vert _ge^{\frac{r}{2}}. \end{aligned}$$

Proof

By the triangle inequality, $\Vert Y_v\Vert _g \leqslant \Vert Z_v\Vert _g + \Vert Y_v-Z_v\Vert _g$, and the result follows from Corollary 4.9 and Proposition 4.11. The same proof applies for $SY_v$. $\square$

4.5 A volume growth upper bound

In this section, we compute a pointwise upper bound on the volume density function $\lambda$ defined in Sect. 3.3 using the normal Jacobi fields estimates derived earlier. The proof relies on an adapted choice for a basis of the tangent space $T_p\partial \!K$ and on Hadamard’s inequality on determinants.

Proposition 4.13

Let (M, g, J) be an (ALCH) manifold of order $a>\frac{1}{2}$ with an essential subset K. Let $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ be the local continuous 1-forms associated to a radially parallel orthonormal frame $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ on a cylinder $E({\mathbb {R}}_+\times U)$. Let $p\in U$ and $k_p$ be the rank of the family $\{\eta |_p,\eta ^1|_p\ldots ,\eta ^{2n}|_p\}$ as linear forms on $T_p\partial \!K$.

• If $\eta |_p=0$, then there exists a constant $\Lambda _+=\Lambda _+(p) >0$ independent of r such that

  $$\begin{aligned} \forall r \geqslant 0, \quad \lambda (r,p) \leqslant \Lambda _+ {\left\{ \begin{array}{ll} e^{(\frac{k_p}{2}-(a-1)(2n+1-k_p))r} &{} \text {if } \frac{1}{2}< a < \frac{3}{2} \\ (r+1)^{2n+1-k_p} e^{(k_p-n-\frac{1}{2})r} &{} \text {if } a = \frac{3}{2}, \\ e^{(k_p-n-\frac{1}{2})r} &{} \text {if } a> \frac{3}{2}. \end{array}\right. } \end{aligned}$$

  (4.5.1)

• If $\eta |_p \ne 0$, then there exists a constant $\Lambda _+ = \Lambda _+(p)>0$ independent of r such that

  $$\begin{aligned} \forall r \geqslant 0, \quad \lambda (r,p) \leqslant \Lambda _+ {\left\{ \begin{array}{ll} e^{(\frac{k_p+1}{2}-(a-1)(2n+1-k_p))r} &{} \text {if } \frac{1}{2}< a < \frac{3}{2} \\ (r+1)^{2n+1-k_p} e^{(k_p-n)r} &{} \text {if } a = \frac{3}{2}, \\ e^{(k_p-n)r} &{} \text {if } a> \frac{3}{2}. \end{array}\right. } \end{aligned}$$

Proof

We first show the case where $\eta |_p = 0$. Without loss of generality, one can assume that $\{\eta ^1|_p,\ldots ,\eta ^{k_p}|_p\}$ generates the family $\{\eta |_p,\eta ^1|_p,\ldots ,\eta ^{2n}|_p\}$. Let $\{v_1,\ldots ,v_{2n+1}\}$ be a basis of $T_p\partial \!K$ such that $\eta ^i|_p(v_j) = \delta ^i_j$ for $(i,j) \in \{1,\ldots ,k_p\}^2$, and $v_{k_p+1},\ldots ,v_{2n+1} \in \cap _{i=1}^{k_p} \ker \eta ^i|_p$. The volume density function is given by the relation

$$\begin{aligned} \forall (r,p) \in {\mathbb {R}}_+\times \partial \!K,\quad \lambda (r,p) = |\det {{\,\textrm{d}\,}}\!E(r,p)|, \end{aligned}$$

where the determinant is taken in any orthonormal bases. It follows that

$$\begin{aligned} \forall r \geqslant 0,\quad \lambda (r,p) = \frac{|\det (\partial _r, Y_1,\ldots , Y_{2n+1})|}{|\det (\nu ,v_1,\ldots ,v_{2n+1})|}, \end{aligned}$$

where $Y_j = Y_{v_j}(\gamma _p(r))$, and all determinants are taken in orthonormal bases. Hadamard’s inequality on determinants now yields

$$\begin{aligned} \forall r \geqslant 0,\quad \lambda (r,p) \leqslant \frac{1}{|\det (\nu ,v_1,\ldots ,v_{2n+1})|}\prod _{i=1}^{2n+1} \Vert Y_i\Vert _g. \end{aligned}$$

According to Proposition 4.11, there exists $C>0$ such that

$$\begin{aligned} \forall i \in \{k_p+1,\ldots ,2n+1\},\quad \Vert Y_i\Vert _g \leqslant C \Vert v_i\Vert _g {\left\{ \begin{array}{ll} e^{-(a-1)r} &{}{} \text{ if } \frac{1}{2}< a < \frac{3}{2}, \\ (r+1)e^{-\frac{r}{2}} &{}{} \text{ if } a = \frac{3}{2}, \\ e^{-\frac{r}{2}} &{}{}\text{ if } a > \frac{3}{2}, \end{array}\right. } \end{aligned}$$

while Lemma 4.12 provides the existence of $c>0$ such that

$$\begin{aligned} \forall i \in \{1,\ldots ,k_p\},\quad \Vert Y_i\Vert _g \leqslant c\Vert v_i\Vert _ge^{\frac{r}{2}}. \end{aligned}$$

The claimed upper bound (4.5.1) now holds with $\Lambda _+ = C^{2n+1-k_p}c^{k_p}\frac{\prod _{i=1}^{2n+1}\Vert v_i\Vert _g}{|\det (v_1,\ldots ,v_{2n+1})|}$.

In case $\eta |_p \ne 0$, the proof is similar. Without loss of generality, one can assume that $\{\eta |_p,\eta ^1|_p,\ldots ,\eta ^{k_p-1}|_p\}$ generates the whole family. In that case, the first vector of the considered basis will have growth of order at most $e^r$, hence the extra $\frac{r}{2}$ term in the exponent of the upper bound. $\square$

4.6 The local coframe

We shall now compare the two bounds on the volume growth given by Propositions 3.7 and 4.13. Note that these two bounds have been derived from considerations on the curvature of different nature.

Proposition 4.14

Let $(M^{2n+2},g,J)$ be an (ALCH) manifold of order $a>1$, with an essential subset $K\subset M$, such that the sectional curvature of $\overline{M\setminus K}$ is non-positive. Then the continuous 1-form $\eta$ is non-vanishing. Furthermore, if $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ is a radially parallel orthonormal frame on a cylinder $E({\mathbb {R}}_+\times U)$, then $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ is a continuous coframe on U.

Proof

Let $p\in \partial \!K$ and let $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame on a cylinder $E({\mathbb {R}}_+\times U)$, with $p\in U \subset \partial \!K$. Let $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ be the continuous local 1-forms on U given by Proposition 4.8, and $k_p$ be the rank of this family as linear forms on $T_p\partial \!K$. Let $\varepsilon \in (0,\min \{a-1,\frac{1}{2}\})$. Propositions 3.7 and 4.13 yield the existence of $r_0=r_0(\varepsilon ,a,C_0)>0$, $\Lambda _-=\Lambda _-(\varepsilon ,a,C_0)>0$ and $\Lambda _+=\Lambda _+(a,C_0,p)>0$ such that

• If $\eta |_p = 0$, then for $r\geqslant r_0$, we get

  $$\begin{aligned} \Lambda _- e^{(n+\frac{1}{2}-\varepsilon )r}\leqslant \lambda (r,p) \leqslant \Lambda _+ {\left\{ \begin{array}{ll} e^{(\frac{k_p}{2}-(a-1)(2n+1-k_p))r} &{} \text {if } 1< a < \frac{3}{2}, \\ (r+1)^{2n+1-k_p} e^{(k_p-n-\frac{1}{2})r} &{} \text {if } a = \frac{3}{2}, \\ e^{(k_p-n-\frac{1}{2})r} &{} \text {if } a> \frac{3}{2}. \end{array}\right. } \end{aligned}$$

  (4.6.1)

• If $\eta |_p \ne 0$, then for $r\geqslant r_0$, we get

  $$\begin{aligned} \Lambda _- e^{(n+\frac{1}{2}-\varepsilon )r}\leqslant \lambda (r,p) \leqslant \Lambda _+ {\left\{ \begin{array}{ll} e^{(\frac{k_p+1}{2}-(a-1)(2n+1-k_p))r} &{} \text {if } 1< a < \frac{3}{2}, \\ (r+1)^{2n+1-k_p} e^{(k_p-n)r} &{} \text {if } a = \frac{3}{2}, \\ e^{(k_p-n)r} &{} \text {if } a> \frac{3}{2}. \end{array}\right. } \end{aligned}$$

  (4.6.2)

By contradiction, assume that $k_p < 2n+1$. Then if $\eta |_p = 0$, a straightforward asymptotic comparison of the lower and upper bounds of (4.6.1) yields

$$\begin{aligned} n+\frac{1}{2}- \varepsilon \leqslant {\left\{ \begin{array}{ll} n - (a-1) &{} \text {if } 1<a<\frac{3}{2}, \\ n - \frac{1}{2} &{} \text {if } a \geqslant \frac{3}{2}, \end{array}\right. } \end{aligned}$$

whereas if $\eta |_p \ne 0$, the same study in (4.6.2) yields

$$\begin{aligned} n+\frac{1}{2} - \varepsilon \leqslant {\left\{ \begin{array}{ll} n +\frac{1}{2}- (a-1) &{} \text {if } 1<a<\frac{3}{2}, \\ n &{} \text {if } a \geqslant \frac{3}{2}. \end{array}\right. } \end{aligned}$$

Since $\varepsilon \in (0,\min \{a-1,\frac{1}{2}\})$, all cases lead to a contradiction. It follows that $k_p = 2n+1$. Thus, $\{\eta |_p,\eta ^1|_p,\ldots ,\eta ^{2n}|_p\}$ is a linearly independent family. This being true for all $p \in U$, $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ is a coframe, and in particular, $\eta$ does not vanish on U. The result follows. $\square$

Remark 4.15

Notice that our technique would not have allowed us to draw a conclusion in the limit case $a=1$ when $\eta |_p\ne 0$.

Corollary 4.16

Under the assumptions of Proposition 4.14, there exists a constant $c>0$ such that

$$\begin{aligned} \forall (p,v)\in T\partial \!K, \forall r \geqslant 0,\quad \Vert Y_v\Vert _g \geqslant c\Vert v\Vert _ge^{\frac{r}{2}}. \end{aligned}$$

Proof

For $r\geqslant 0$, we define a field of quadratic forms $Q_r$ on $\partial \!K$ by the relation

$$\begin{aligned} Q_r(v) = e^{-2r}g(Y_v,J\!\partial _r)^2 + e^{-r}\Vert Y_v^{\perp }\Vert _g^2, \end{aligned}$$

where $Y_v^{\perp }$ denotes the orthogonal projection of $Y_v$ onto $\{\partial _r,J\!\partial _r\}^{\perp }$. In a radially parallel orthonormal frame, $Q_r$ reads $Q_r = {\eta _r}^2 + \sum _{j=1}^{2n}{(\eta ^j_r)}^2$. It follows from Proposition 4.8 that $Q_r$ locally uniformly converges on $\partial \!K$, to a field of quadratic forms $Q_{\infty }$. Since $\partial \!K$ is compact, the convergence is uniform. The limit locally reads $Q_{\infty } = \eta \otimes \eta + \sum _{j=1}^{2n} \eta ^j\otimes \eta ^j$, which is then positive definite by Proposition 4.14. Let $U\partial \!K\subset T\partial \!K$ be the unit sphere bundle. Then $(r,u)\in [0,+\infty ]\times U\partial \!K\mapsto Q_r(u)$ is continuous on the compact set $[0,+\infty ]\times U\partial \!K$ and achieves its minimum at a point $(r_{\min },u_{\min })$. Since $Q_{r_{\min }}$ is positive definite, this minimum is positive. It follows by homogeneity that for any tangent vector v on $\partial \!K$, it holds

$$\begin{aligned} \Vert Y_v\Vert _g^2 \geqslant e^r Q_r(v) \geqslant e^r \Vert v\Vert _g^2Q_{r_{\min }}(u_{\min }). \end{aligned}$$

The result then follows. $\square$

In particular, if $(g_r)_{r\geqslant 0}$ denotes the family of Riemannian metrics induced by the normal exponential chart on $\partial \!K$ and if (M, g, J) is (ALCH) of order $a>1$, then there exists $c,C>0$ such that $ce^r g_0 \leqslant g_r \leqslant Ce^{2r}g_0$.

4.7 Asymptotic development of the metric

We are now able to state again our first Theorem.

Theorem A

Let $(M^{2n+2},g,J)$ be an (ALCH) manifold of order $a>1$ with an essential subset K, such that the sectional curvature of $\overline{M\setminus K}$ is negative. Then there exist a continuous non-vanishing 1-form $\eta$ and a continuous Carnot-Carathéodory metric $\gamma _H$ on $\partial \!K$, such that

$$\begin{aligned} E^*g = {{\,\textrm{d}\,}}\!r^2 + e^{2r} \eta \otimes \eta + e^r \gamma _H+ h_r, \end{aligned}$$

where $(h_r)_{r\geqslant 0}$ is a smooth family of symmetric bilinear forms on $\partial \!K$ such that there exists $C>0$ with

$$\begin{aligned} \forall r \geqslant 0, \forall u,v,\quad |h_r(u,v)| \leqslant C\Vert u\Vert _g\Vert v\Vert _g {\left\{ \begin{array}{ll} e^{(2-a)r} &{} \text {if } 1<a<\frac{3}{2}, \\ (r+1)e^{\frac{r}{2}} &{} \text {if } a = \frac{3}{2},\\ e^{\frac{r}{2}} &{} \text {if } a > \frac{3}{2}. \end{array}\right. } \end{aligned}$$

Moreover, $\gamma _H$ is a continuous positive-definite metric on $H=\ker \eta$.

Proof

Let us first notice that Gauss Lemma yields the existence of a smooth family of metrics $(g_r)_{r\geqslant 0}$ on $\partial \!K$ such that in the normal exponential chart, the metric reads $E^*g = {{\,\textrm{d}\,}}\!r ^2 + g_r$. Let us fix $p \in \partial \!K$. Let $U\subset \partial \!K$ be an open neighbourhood of p on which is defined the local coframe $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ given by Proposition 4.14. Let u and v be two tangent vectors at p.

First, notice that $g_r(u,v) = g(Y_u,Y_v)$, where $Y_u$ and $Y_v$ are defined in Sect. 2.1. Second, notice that

$$\begin{aligned} g(Z_u,Z_v) = e^{2r}\eta (u)\eta (v)+e^r\sum _{j=1}^{2n}\eta ^j(u)\eta ^j(v), \end{aligned}$$

where $Z_u$ and $Z_v$ are defined in Definition 4.10. Define $h_r$ by the relation

$$\begin{aligned} h_r(u,v) = g(Z_u,Y_v-Z_v) + g(Y_u-Z_u,Z_v)+g(Y_u-Z_u,Y_v-Z_v), \end{aligned}$$

It follows from the triangle inequality, Cauchy-Schwarz inequality, Proposition 4.11 and Lemma 4.12, that there exists a constant $C>0$ independent of r, p, u and v such that

$$\begin{aligned} |h_r(u,v)| \leqslant C\Vert u\Vert _g\Vert v\Vert _g {\left\{ \begin{array}{ll} e^{(2-a)r} &{} \text {if } 1<a<\frac{3}{2}, \\ (r+1)e^{\frac{r}{2}} &{} \text {if } a = \frac{3}{2},\\ e^{\frac{r}{2}} &{} \text {if } a > \frac{3}{2}. \end{array}\right. } \end{aligned}$$

(4.7.1)

Let $\gamma _H$ be defined by the relation $\gamma _H = \lim _{r\rightarrow +\infty } e^{-r}\left( g_r - e^{2r}\eta \otimes \eta \right)$. Notice that

$$\begin{aligned} g_r = e^{2r}\eta \otimes \eta + e^r \sum _{j=1}^{2n} \eta ^j\otimes \eta ^j + h_r, \end{aligned}$$

so that it follows from (4.7.1) that $\gamma _H$ locally reads $\gamma _H = \sum _{j=1}^{2n} \eta ^j\otimes \eta ^j$, and $\gamma _H$ is hence continuous and positive semi-definite. Since $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ is a local coframe, $\gamma _H$ is thus non-degenerate on $H = \ker \eta$. This concludes the proof. $\square$

Corollary 4.17

Under the assumptions of Theorem A, there exists a unique continuous vector field $\xi$ on $\partial \!K$ such that

• 1. $\eta (\xi ) = 1$,

• 2. $\gamma _H(\xi ,\cdot ) = 0$.

Proof

Let us locally define the vector field $\xi$ as the first vector of the dual basis of the local coframe $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$. Then $\xi$ is uniquely characterized by 1. and 2., so that $\xi$ does not depend on the choice of the radially parallel orthonormal frame defining the coframe, and is defined on all of $\partial \!K$. Since the local coframe is continuous, so is $\xi$. This concludes the proof. $\square$

Definition 4.18

(Canonical elements at infinity) Under the assumptions of Theorem A, we call $\eta$ the canonical 1-form at infinity, $H = \ker \eta$ the canonical distribution at infinity, $\gamma _H$ the canonical Carnot-Carathéodory metric at infinity and $\xi$ the canonical vector field at infinity.

We call these elements canonical because they are global and do not depend on the choice of any radially parallel orthonormal frame. The analogy with the model case is obvious: $\eta$ corresponds to the contact form $\theta$, $\gamma _H$ corresponds to the Levi form $\gamma$ and $\xi$ corresponds to the Reeb vector field of $\theta$. Notice that no assumption on $\nabla R$ has been made yet. In the following section, we shall pursue the study of this analogy, and show that under stronger assumptions on R and $\nabla R$, the canonical elements at infinity have ${\mathcal {C}}^1$ regularity, and that $\eta$ is indeed a contact form.

5 The contact structure

Let (M, g, J) be an (ALCH) manifold of order $a>1$, with an essential subset K, such that the sectional curvature of $\overline{M{\setminus } K}$ is non-positive. We aim to show that the canonical 1-form $\eta$ is a contact form of class ${\mathcal {C}}^1$. To do so, we show that the family of 1-forms $\{\eta _r\}_{r\geqslant 0}$ defined in Definition 4.3 converges to $\eta$ in ${\mathcal {C}}^1$ topology.

Let us first show the following Lemma.

Lemma 5.1

If u and v are vector fields on $\partial \!K$, then

$$\begin{aligned} \forall r \geqslant 0, \quad ({\mathcal {L}}_u\eta _r)(v) = e^{-r}\big ( g(\nabla _{Y_v}Y_u,J\!\partial _r) + g(Y_v,\nabla _{Y_u}J\!\partial _r)\big ). \end{aligned}$$

Proof

By basic properties of the Lie derivative, it holds that

$$\begin{aligned} ({\mathcal {L}}_u\eta _r)(v) = u \cdot \eta _r(v) - \eta _r([u,v]). \end{aligned}$$

Hence, the equalities $Y_u=E_*u$ and $Y_v = E_*v$ together with the naturality of the Lie bracket yield

$$\begin{aligned} ({\mathcal {L}}_u\eta _r)(v) = Y_u \cdot e^{-r}g(Y_v,J\!\partial _r) - e^r g([Y_u,Y_v],J\!\partial _r). \end{aligned}$$

Recall that $Y_u$ and $\partial _r$ are orthogonal. Since $\partial _r$ is the gradient of the distance function r, it follows that $Y_u\cdot e^rg(Y_v,J\!\partial _r) = e^r Y_u\cdot g(Y_v,J\!\partial _r)$. The torsion-free property of the Levi-Civita connection concludes the proof. $\square$

Computations show that in order to estimate the asymptotic behaviour of the Lie derivative $({\mathcal {L}}_u\eta _r)(v)$, an upper-bound on the norm of $\nabla _{Y_v}Y_u$ is needed. This upper bound in itself is obtained by controlling the norm of $(\nabla _{Y_v}S)Y_u$. The following subsection takes on the tedious task of providing such upper bounds.

5.1 Order one estimates

In this subsection, u and v are fixed vector fields on $\partial \!K$.

Lemma 5.2

The following holds

$$\begin{aligned} \begin{aligned} \frac{1}{2}\partial _r\Vert (\nabla _{Y_v}S)Y_u\Vert _g^2&= R(\partial _r,Y_v,SY_u,S(\nabla _{Y_v}S)Y_u) - R(\partial _r,SY_v,Y_u,(\nabla _{Y_v}S)Y_u)\\&\quad - R(SY_v,Y_u,\partial _r,(\nabla _{Y_v}S)Y_u) - R(\partial _r,Y_u,SY_v,(\nabla _{Y_v}S)Y_u) \\&\quad - (\nabla _{Y_v}R)(\partial _r,Y_u,\partial _r,(\nabla _{Y_v}S)Y_u) - g(S(\nabla _{Y_v}S)Y_u,(\nabla _{Y_v}S)Y_u). \end{aligned} \end{aligned}$$

Proof

The extension of the covariant derivative to the whole tensor algebra gives the equality $(\nabla _{Y_v}S)Y_u = \nabla _{Y_v}(SY_u) - S \nabla _{Y_v}Y_u$. It then follows that

$$\begin{aligned} \nabla _{\partial _r}((\nabla _{Y_v}S)Y_u)) = \nabla _{\partial _r}\big ( \nabla _{Y_v}(SY_u)\big ) - (\nabla _{\partial _r}S)\nabla _{Y_v}Y_u - S\nabla _{\partial _r}(\nabla _{Y_v}Y_u). \end{aligned}$$

(5.1.1)

From Lemma 2.1, $[\partial _r,Y_v]=0$, $\nabla _{\partial _r}Y_u = SY_u$ and $\nabla _{\partial _r}(SY_u) = -R_{\partial _r}Y_u$. It follows from our convention on the Riemann curvature tensor (see (2.1.1)) that

$$\begin{aligned} \begin{aligned} \nabla _{\partial _r}\big ( \nabla _{Y_v}(SY_u)\big )&= -R(\partial _r,Y_v)SY_u + \nabla _{Y_v}(-R_{\partial _r}Y_u) \\&= -R(\partial _r,Y_v)SY_u - (\nabla _{Y_v}R)(\partial _r,Y_u)\partial _r\\&\quad - R(SY_v,Y_u)\partial _r- R(\partial _r,\nabla _{Y_v}Y_u)\partial _r\\&\quad - R(\partial _r,Y_u)SY_v, \end{aligned} \end{aligned}$$

(5.1.2)

as well as

$$\begin{aligned} \begin{aligned} S\nabla _{\partial _r}(\nabla _{Y_v}Y_u)&= -SR(\partial _r,Y_v)Y_u + S\nabla _{Y_v}(SY_u)\\&= -SR(\partial _r,Y_v)Y_u + S^2\nabla _{Y_v}Y_u + S\big (\nabla _{Y_v}S)Y_u\big ). \end{aligned} \end{aligned}$$

(5.1.3)

The Riccati equation (2.1.2) for S yields

$$\begin{aligned} (\nabla _{\partial _r}S) \nabla _{Y_v}Y_u = -S^2 \nabla _{Y_y}Y_u - R(\partial _r,\nabla _{Y_v}Y_u)\partial _r. \end{aligned}$$

(5.1.4)

Inserting (5.1.2), (5.1.3) and (5.1.4) into (5.1.1) now gives

$$\begin{aligned} \begin{aligned} \nabla _{\partial _r}((\nabla _{Y_v}S)Y_u))&= SR(\partial _r,Y_v)Y_u - R(\partial _r,Y_v)SY_u\\&\quad - R(SY_v,Y_u)\partial _r- R(\partial _r,Y_u)SY_v\\&\quad - (\nabla _{Y_v}R)(\partial _r,Y_u)\partial _r- S(\nabla _{Y_v}S)Y_u. \end{aligned} \end{aligned}$$

The result now follows from the symmetry of S and the equality

$$\begin{aligned} \frac{1}{2}\partial _r\Vert (\nabla _{Y_v}S)Y_u\Vert _g^2 = g(\nabla _{\partial _r}((\nabla _{Y_v}S)Y_u)),(\nabla _{Y_v}S)Y_u). \end{aligned}$$

$\square$

Proposition 5.3

If in addition (M, g, J) has (ALS) property of order $b>0$, then there exists $c>0$ such that

$$\begin{aligned} \Vert (\nabla _{Y_v}S)Y_u\Vert _g \leqslant c \Vert u\Vert _g\Vert v\Vert _ge^{2r}. \end{aligned}$$

Proof

Let $p\in \partial \!K$. Define $F(r) = \Vert ((\nabla _{Y_v}S)Y_u)({\gamma _p(r)})\Vert _g$. Since the norm is Lipschitz and the entries are smooth, F is locally Lipschitz. It follows from Rademacher’s Theorem that F is almost everywhere differentiable, and Lemma 5.2 yields the almost everywhere equality

$$\begin{aligned} \begin{aligned} F' F&= R(\partial _r,Y_v,SY_u,S(\nabla _{Y_v}S)Y_u) - R(\partial _r,SY_v,Y_u,(\nabla _{Y_v}S)Y_u)\\&\quad - R(SY_v,Y_u,\partial _r,(\nabla _{Y_v}S)Y_u) - R(\partial _r,Y_u,SY_v,(\nabla _{Y_v}S)Y_u) \\&\quad - (\nabla _{Y_v}R)(\partial _r,Y_u,\partial _r,(\nabla _{Y_v}S)Y_u) - g(S(\nabla _{Y_v}S)Y_u,(\nabla _{Y_v}S)Y_u). \end{aligned} \end{aligned}$$

Since S is positive, $g(S(\nabla _{Y_v}S)Y_u,(\nabla _{Y_v}S)Y_u) \geqslant 0$. Therefore, it follows that

$$\begin{aligned} F'F \leqslant + (4\Vert R\Vert _g \Vert S\Vert _g + \Vert \nabla R\Vert _g) \Vert Y_v\Vert _g\Vert Y_u\Vert _gF \quad a.e. \end{aligned}$$

Recall that $\Vert R\Vert _g$ (Lemma 3.3), $\Vert \nabla R\Vert _g$ ((ALS) assumption) and $\Vert S\Vert _g$ (Lemma 3.5) are uniformly bounded. In addition, recall that there exists a constant $c_1>0$ such that $\Vert Y_u\Vert _g\leqslant c_1\Vert u\Vert _ge^r$ and $\Vert Y_v\Vert _g\leqslant c_1\Vert v\Vert _ge^r$ (Lemma 4.12). Hence, there exists a constant $c>0$ such that

$$\begin{aligned} F'F \leqslant c\Vert u\Vert _g\Vert v\Vert _ge^{2r} F \quad a.e, \end{aligned}$$

and therefore,

$$\begin{aligned} F' \leqslant c\Vert u\Vert _g\Vert v\Vert _g e^{2r} \quad a.e, \end{aligned}$$

which is true even at points where F vanishes since at those points, it achieves a minimum and its derivative thus vanishes. The result now follows from a straightforward integration. $\square$

Remark 5.4

This bound is sharp, since $g((\nabla _{Y_v}S)Y_u,\partial _r) = -g(SY_v,SY_u)$ is equivalent to $-\eta (u)\eta (v)e^{2r}$. It is however worth noticing that it can be proven that the normal component $(\nabla _{Y_v}S)Y_u^{\perp }=(\nabla _{Y_v}S)Y_u - g((\nabla _{Y_v}S)Y_u,\partial _r)\partial _r$ can be bounded, if $\min \{a,b\}>\frac{1}{2}$, as follows

$$\begin{aligned} \Vert (\nabla _{Y_v}S)Y_u^{\perp }\Vert _g \leqslant {\widetilde{c}}\Vert u\Vert _g\Vert v\Vert _g e^{\frac{3}{2}r}. \end{aligned}$$

The proof is more intricate, and relies on the study of the curvature terms. For example, the terms in $R^0(\partial _r,Y_v,SY_u,S(\nabla _{Y_v}S)Y_u)$ are products of the form $g(X,J\!\partial _r)g(Y,JZ)$ with X, Y, Z normal to $\partial _r$, which kills the $\partial _r$ direction.

We shall now give a bound on the growth of $\nabla _{Y_v}Y_u$. The proof is slightly different from that of Proposition 5.3 as it requires the use of Grönwall’s inequality.

Proposition 5.5

Assume that $a>\frac{1}{2}$ and $b>0$. Then there exists a constant $c>0$ such that

$$\begin{aligned} \Vert (\nabla _{Y_v}Y_u)(\gamma _p(r))\Vert _g \leqslant c(\Vert (\nabla _{Y_v}Y_u)(p)\Vert _g+ \Vert u\Vert _g\Vert v\Vert _g) e^{2r}. \end{aligned}$$

Proof

First, notice that $\nabla _{\partial _r}(\nabla _{Y_v}Y_u) = -R(\partial _r,Y_v)Y_u + \nabla _{Y_v}(SY_u)$ and moreover that $\nabla _{Y_v}(SY_u) = (\nabla _{Y_v}S)Y_u + S\nabla _{Y_v}Y_u$. Hence

$$\begin{aligned} \begin{aligned} \frac{1}{2}\partial _r\Vert \nabla _{Y_v}Y_u\Vert _g^2&= -R(\partial _r,Y_v,Y_u,\nabla _{Y_v}Y_u) + g((\nabla _{Y_v}S)Y_u,\nabla _{Y_v}Y_u) \\&\quad +g(S\nabla _{Y_v}Y_u,\nabla _{Y_v}Y_u). \end{aligned} \end{aligned}$$

(5.1.5)

Let $p\in \partial \!K$ be fixed and let $G(r) = \Vert (\nabla _{Y_v}Y_u)({\gamma _p(r)})\Vert _g$. Then G is almost everywhere differentiable and the triangle inequality together with Cauchy-Schwarz inequality applied to (5.1.5) yields the almost everywhere inequality

$$\begin{aligned} G'G \leqslant \Vert R\Vert _g\Vert Y_u\Vert _g\Vert Y_v\Vert _gG + \Vert (\nabla _{Y_v}S)Y_u\Vert _gG + \Vert S\Vert _gG^2\quad a.e, \end{aligned}$$

from which is deduced the following almost everywhere inequality

$$\begin{aligned} G' \leqslant \Vert R\Vert _g\Vert Y_u\Vert _g\Vert Y_v\Vert _g + \Vert (\nabla _{Y_v}S)Y_u\Vert _g + \Vert S\Vert _gG \quad a.e, \end{aligned}$$

which is true even at points where G vanishes since at those points, it achieves a minimum and its derivative thus vanishes. Estimates on the growth of normal Jacobi fields (Lemma 4.12), a uniform bound on $\Vert R\Vert _g$ (Lemma 3.3) together with Proposition 5.3 give the existence of a constant $C>0$ such that

$$\begin{aligned} G' - G \leqslant C\Vert u\Vert _g\Vert v\Vert _ge^{2r} + (\Vert S\Vert _g-1)G \quad a.e. \end{aligned}$$

Multiplying both sides by $e^{-r}$ and integrating gives the Grönwall-like inequality

$$\begin{aligned} \forall r \geqslant 0,\quad G(r) e^{-r} \leqslant G(0) +C\Vert u\Vert _g\Vert v\Vert _g (e^r-1) + \int _0^r (\Vert S\Vert _g-1)(G(s)e^{-s}){{\,\textrm{d}\,}}\!s. \end{aligned}$$

Finally, recall from Proposition 3.5 that

$$\begin{aligned} \Vert S\Vert _g - 1 \leqslant \epsilon _a(r)=C' {\left\{ \begin{array}{ll} e^{-ar} &{} \text {if } \frac{1}{2}< a < 2, \\ (r+1)e^{-2r} &{} \text {if } a = 2,\\ e^{-2r} &{} \text {if } a >2, \end{array}\right. } \end{aligned}$$

for some constant $C'$ independent of (r, p, u, v). Hence, Grönwall’s inequality yields

$$\begin{aligned} \forall r \geqslant 0,\quad G(r)e^{-r} \leqslant (G(0) + C\Vert u\Vert _g\Vert v\Vert _ge^r)\exp \left( \int _0^r \epsilon _a(s){{\,\textrm{d}\,}}\!s\right) . \end{aligned}$$

It follows that there exists $c>0$ such that

$$\begin{aligned} \Vert (\nabla _{Y_v}Y_u)(\gamma _p(r))\Vert _g \leqslant c(\Vert (\nabla _{Y_v}Y_u)(p)\Vert _g + \Vert u\Vert _g\Vert v\Vert _g)e^{2r}. \end{aligned}$$

$\square$

Remark 5.6

• This bound is sharp since $g(\nabla _{Y_v}Y_u,\partial _r) = - g(SY_v,Y_u)$ is equivalent to $-\eta (u)\eta (v)e^{2r}$. Similarly to Proposition 5.3, it is possible to give a sharper bound on the normal component, of order $e^{\frac{3}{2}r}$.

• Since $\nabla _{Y_v}Y_u$ is not tensorial in u, one cannot hope to find a constant $c>0$ independent of u such that $\Vert \nabla _{Y_v}Y_u\Vert _g \leqslant c \Vert u\Vert _g\Vert v\Vert _ge^{2r}$. However, when u and v are fixed vector fields, the constant in front of the exponential term is continuous with respect to p, and hence uniformly bounded on all compact subsets on which it is defined.

5.2 The contact form

From now on, we assume that (M, g, J) is (ALCH) and (ALS) of orders $a>1$ and $b>0$, and also that the sectional curvature of $\overline{M\setminus K}$ is negative. In this section, we prove that the canonical 1-form at infinity is of class ${\mathcal {C}}^1$ and is contact. We first show the following computational Proposition.

Proposition 5.7

Assume that $\min \{a,b\}>1$. Let u and v be local tangent vector fields on $\partial \!K$. Define

$$\begin{aligned} f(r,p) = g(\nabla _{Y_v}Y_u,J\!\partial _r) + g(\nabla _{Y_u}J\!\partial _r,Y_v). \end{aligned}$$

Then there exists a constant $c>0$ independent of (r, p, u, v) and a continuous function $\alpha (p)$ such that

$$\begin{aligned} |f(r,p)-{e^r}\alpha (p) |\leqslant c(\Vert u\Vert _g\Vert v\Vert _g + \Vert \nabla _{Y_v}Y_u\Vert _g){\left\{ \begin{array}{ll} e^{(2-\min \{a,b\})r} &{} \text {if } 1<\min \{a,b\} < 3,\\ (r+1)e^{-r} &{} \text {if } \min \{a,b\} ={3}, \\ e^{-r} &{} \text {if } \min \{a,b\}>3. \end{array}\right. } \end{aligned}$$

Proof

Our convention on the curvature yields $\nabla _{\partial _r}(\nabla _{Y_v}Y_u) = -R(\partial _r,Y_v)Y_u+ \nabla _{Y_v}(SY_u)$ and $\nabla _{\partial _r}(\nabla _{Y_u}J\!\partial _r) = -R(\partial _r,Y_u)J\!\partial _r$. Therefore,

$$\begin{aligned} \begin{aligned} \partial _rf&= -R(\partial _r,Y_v,Y_u,J\!\partial _r) + g(\nabla _{Y_v}(SY_u),J\!\partial _r)\\&\quad -R(\partial _r,Y_u,J\!\partial _r,Y_v) + g(\nabla _{Y_u}J\!\partial _r,SY_v), \end{aligned} \end{aligned}$$

which turns out, by the Bianchi identity, to be equal to the following

$$\begin{aligned} \partial _rf = R(\partial _r,J\!\partial _r,Y_v,Y_u) + g(\nabla _{Y_v}(SY_u),J\!\partial _r) + g(\nabla _{Y_u}J\!\partial _r,SY_v). \end{aligned}$$

(5.2.1)

Notice that

$$\begin{aligned} \begin{aligned} \nabla _{\partial _r}(\nabla _{Y_v}(SY_u))&= -R(\partial _r,Y_v)SY_u + \nabla _{Y_v}(\nabla _{\partial _r}(SY_u))\\&= -R(\partial _r,Y_v)SY_u + \nabla _{Y_v}(-R_{\partial _r}Y_u)\\&= -R(\partial _r,Y_v)SY_u - (\nabla _{Y_v}R)(\partial _r,Y_u)\partial _r- R(SY_v,Y_u)\partial _r\\&\quad - R(\partial _r,\nabla _{Y_v}Y_u)\partial _r- R(\partial _r,Y_u)SY_v. \end{aligned} \end{aligned}$$

It then follows that

$$\begin{aligned} \begin{aligned} \partial _r\partial _rf&= (\nabla _{\partial _r}R)(\partial _r,J\!\partial _r,Y_v,Y_u) + R(\partial _r,J\!\partial _r,SY_v,Y_u) \\&\quad + R(\partial _r,J\!\partial _r,Y_v,SY_u) -R(\partial _r,Y_v,SY_u,J\!\partial _r)\\&\quad - (\nabla _{Y_v}R)(\partial _r,Y_u,\partial _r,J\!\partial _r) - R(SY_v,Y_u,\partial _r,J\!\partial _r)\\&\quad -R(\partial _r,\nabla _{Y_v}Y_u,\partial _r,J\!\partial _r) -R(\partial _r,Y_u,SY_v,J\!\partial _r) \\&\quad -R(\partial _r,Y_u,J\!\partial _r,SY_v) -R(\partial _r,Y_v,\partial _r,\nabla _{Y_u}J\!\partial _r). \end{aligned} \end{aligned}$$

(5.2.2)

Note that $R(\partial _r,J\!\partial _r,SY_v,Y_u) - R(SY_v,Y_u,\partial _r,J\!\partial _r)=0$ because of the symmetry of R. Similarly, notice that $-R(\partial _r,Y_u,SY_v,J\!\partial _r) - R(\partial _r,Y_u,J\!\partial _r,SY_v) = 0$. Hence, equation (5.2.2) becomes

$$\begin{aligned} \begin{aligned} \partial _r\partial _rf&= (\nabla _{\partial _r}R) (\partial _r,J\!\partial _r,Y_v,Y_u) - (\nabla _{Y_v} R) (\partial _r,Y_u,\partial _r,J\!\partial _r)\\&\quad + R(\partial _r,J\!\partial _r,Y_v,SY_u) - R(\partial _r,Y_v,SY_u,J\!\partial _r)\\&\quad -R(\partial _r,\nabla _{Y_v}Y_u,\partial _r,J\!\partial _r) - R(\partial _r,Y_v,\partial _r,\nabla _{Y_u}J\!\partial _r). \end{aligned} \end{aligned}$$

Let $\mathbf {k^0}$ be defined as

$$\begin{aligned} \begin{aligned} \mathbf {k^0}&= R^0(\partial _r,J\!\partial _r,Y_v,SY_u) - R^0(\partial _r,Y_v,SY_u,J\!\partial _r)\\&\quad -R^0(\partial _r,\nabla _{Y_v}Y_u,\partial _r,J\!\partial _r) - R^0(\partial _r,Y_v,\partial _r,\nabla _{Y_u}J\!\partial _r). \end{aligned} \end{aligned}$$

From Lemma A.1 in the Appendix, it holds that

$$\begin{aligned} \begin{aligned} {\textbf{k}}^0&= -\frac{1}{2}g(SY_u,JY_v) -\frac{1}{4}g(SY_u,JY_v) + g(\nabla _{Y_v}Y_u,J\!\partial _r) + \frac{1}{4}g(Y_v,\nabla _{Y_u}J\!\partial _r)\\&= -\frac{3}{4}g(SY_u,JY_v) + \frac{1}{4}g(Y_v,\nabla _{Y_u}J\!\partial _r) + g(\nabla _{Y_v}Y_u,J\!\partial _r). \end{aligned} \end{aligned}$$

Note that $g(SY_u,JY_v) = g(\nabla _{Y_u}\partial _r,JY_v) = -g(J\nabla _{Y_u}\partial _r,Y_v)=-g(\nabla _{Y_u}J\!\partial _r,Y_v)$. It surprisingly turns out that

$$\begin{aligned} {\textbf{k}}^0 = g(\nabla _{Y_v}Y_u,J\!\partial _r) + g(\nabla _{Y_u}J\!\partial _r,J\!\partial _r) = f. \end{aligned}$$

Hence, f a is solution to the second order linear differential equation

$$\begin{aligned} \partial _r\partial _rf - f = h, \end{aligned}$$

where h is given by

$$\begin{aligned} \begin{aligned} h&= (\nabla _{\partial _r}R) (\partial _r,J\!\partial _r,Y_v,Y_u) - (\nabla _{Y_v} R) (\partial _r,Y_u,\partial _r,J\!\partial _r)\\&\quad + (R-R^0)(\partial _r,J\!\partial _r,Y_v,SY_u) - (R-R^0)(\partial _r,Y_v,SY_u,J\!\partial _r)\\&\quad -(R-R^0)(\partial _r,\nabla _{Y_v}Y_u,\partial _r,J\!\partial _r) - (R-R^0)(\partial _r,Y_v,\partial _r,\nabla _{Y_u}J\!\partial _r). \end{aligned} \end{aligned}$$

(5.2.3)

By classical ODE theory, f reads

$$\begin{aligned} f = \frac{e^r}{2}\big ({f(0)+\partial _rf(0)} +\int _0^re^{-s}h(s){{\,\textrm{d}\,}}\!s\big ) +\frac{e^{-r}}{2}\big (f(0) -\partial _rf(0) - \int _0^r e^s h(s) {{\,\textrm{d}\,}}\!s \big ). \end{aligned}$$

Since $\nabla _{Y_u}J\!\partial _r= JSY_u$, it holds that $\Vert JSY_u\Vert _g = \Vert SY_u\Vert _g \leqslant \Vert S\Vert _g\Vert Y_u\Vert _g$, and Eq. (5.2.3) yields the following upper bound on h

$$\begin{aligned} |h| \leqslant (3\Vert R-R^0\Vert _g\Vert S\Vert _g + 2\Vert \nabla R\Vert _g)\Vert Y_u\Vert _g\Vert Y_v\Vert _g + \Vert R-R^0\Vert _g\Vert \nabla _{Y_v}Y_u\Vert _g. \end{aligned}$$

From the normal Jacobi estimates, the uniform bound on $\Vert S\Vert _g$, the (ALCH) and (ALS) conditions and Proposition 5.5, it finally holds that there exists $c>0$ such that

$$\begin{aligned} |h| \leqslant c(\Vert u\Vert _g\Vert v\Vert _g + \Vert (\nabla _{Y_v}Y_u)(p)\Vert _g) e^{(2-\min \{a,b\})r}. \end{aligned}$$

(5.2.4)

Let $\alpha (p)$ be defined as $\alpha (p) = \frac{1}{2}\big (f(0,p) + \partial _rf(0,p) + \int _0^{+\infty } e^{-s}h(s,p){{\,\textrm{d}\,}}\!s\big )$, which is well defined by (5.2.4), and is continuous by the dominated convergence theorem applied on all compact subset where u and v are defined. Hence,

$$\begin{aligned} \begin{aligned} |f(r,p)-\alpha (p)e^r|&\leqslant \frac{e^r}{2}\int _r^{+\infty }e^{-s}|h(s,p)|{{\,\textrm{d}\,}}\!s \\&\quad + \frac{e^{-r}}{2}(|f(0,p)| + |\partial _rf(0,p)| + \int _0^r e^{s}|h(s,p)| {{\,\textrm{d}\,}}\!s). \end{aligned} \end{aligned}$$

Finally, notice that $|f(0,p)| = |g((\nabla _{Y_v}Y_u)(p),J\!\partial _r)| \leqslant \Vert (\nabla _{Y_v}Y_u)(p)\Vert _g$. It now follows from (5.2.1) that

$$\begin{aligned} \begin{aligned} |\partial _rf(0,p)|&= | R(\nu (p),J\nu (p),v,u) + g(((\nabla _{Y_v}S)_p) u,J\!\partial _r)\\&\quad + g(S(\nabla _{Y_v}Y_u)(p),J\!\partial _r) + g(JSu,Sv)|\\&\leqslant \Vert R\Vert _g\Vert v\Vert _g\Vert u\Vert _g + \Vert (\nabla S)_p\Vert _g\Vert u\Vert _g\Vert v\Vert _g \\&\quad + \Vert S\Vert _g\Vert (\nabla _{Y_v}Y_u)(p)\Vert _g + \Vert S\Vert _g\Vert u\Vert _g\Vert v\Vert _g. \end{aligned} \end{aligned}$$

The quantities $\Vert R\Vert _g$, $\Vert S\Vert _g$, and $\Vert \nabla S\Vert _g$ are uniformly bounded on the compact set $\partial \!K$. Moreover, (5.2.4) yields

$$\begin{aligned} \int _r^{+\infty } e^{-s}|h(s,p)|{{\,\textrm{d}\,}}\!s \leqslant c(\Vert u\Vert _g\Vert v\Vert _g + \Vert (\nabla _{Y_v}Y_u)(p)\Vert _g) \frac{e^{(1-\min \{a,b\})r}}{\min \{a,b\}-1}, \end{aligned}$$

and

$$\begin{aligned} \int _0^r e^s |h(s,p)|{{\,\textrm{d}\,}}\!s \leqslant c(\Vert u\Vert _g\Vert v\Vert _g + \Vert (\nabla _{Y_v}Y_u)(p)\Vert _g) {\left\{ \begin{array}{ll} \frac{e^{(3-\min \{a,b\})r}-1}{3-\min \{a,b\}} &{} \text {if } \min \{a,b\} \ne 3, \\ r &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

The result then follows. $\square$

We shall now prove our second Theorem, which we restate for the reader’s convenience.

Theorem B

Let (M, g, J) be an (ALCH) and (ALS) manifold of order a and b, with an essential subset K, such that the sectional curvature of $\overline{M\setminus K}$ is negative. If $\min \{a,b\}>1$, then the canonical 1-form at infinity $\eta$ is a contact form of class ${\mathcal {C}}^1$, with Reeb vector field $\xi$.

Proof

To show that $\eta$ is of class ${\mathcal {C}}^1$, it suffices to fix a chart $U\subset \partial \!K$ and to show that $\eta$ is of class ${\mathcal {C}}^1$ on U. Let $m = 2n+1 = \dim \partial \!K$, $\{x^1,\ldots ,x^{m}\}$ be coordinates on $U\subset \partial \!K$, and let $\{\partial _1,\ldots ,\partial _{m}\}$ and $\{{{\,\textrm{d}\,}}\!x^1,\ldots , {{\,\textrm{d}\,}}\!x^{m}\}$ be the associated tangent frame and coframe. For $r\geqslant 0$, the 1-form $\eta _r$ and its partial derivatives locally read

$$\begin{aligned} \eta _r = \sum _{j=1}^{m} (\eta _r)_j {{\,\textrm{d}\,}}\!x^j \quad \text {and} \quad \partial _i (\eta _r) = \sum _{i=1}^{m} \partial _i (\eta _r)_j {{\,\textrm{d}\,}}\!x^j, \end{aligned}$$

with $\partial _i (\eta _r)_j = ({\mathcal {L}}_{\partial _i} \eta _r)(\partial _j)$. Write $Y_i = Y_{\partial _i}$ and $Y_j= Y_{\partial _j}$. Lemma 5.1 then states that

$$\begin{aligned} \forall i,j\in \{1,\ldots ,m\},\quad \partial _i(\eta _r)_j = e^{-r}\big ( g(\nabla _{Y_j}Y_i,J\!\partial _r) + g(\nabla _{Y_i}J\!\partial _r,Y_j)\big ). \end{aligned}$$

Proposition 5.7 now yields the existence of continuous functions $\alpha _{ij}:U \rightarrow {\mathbb {R}}$ and a constant $c>0$ such that if $i,j \in \{1,\ldots ,m\}$, then

$$\begin{aligned} |\partial _i(\eta _r)_j - \alpha _{ij}| \leqslant c(\Vert \partial _i\Vert _g\Vert \partial _j\Vert _g + \Vert \nabla _{\partial _j}\partial _i\Vert _g ) {\left\{ \begin{array}{ll} e^{(1-\min \{a,b\})r} &{} \text {if } 1<\min \{a,b\} < 3,\\ (r+1)e^{-2r} &{} \text {if } \min \{a,b\} = 3,\\ e^{-2r} &{} \text {if } \min \{a,b\} >3. \end{array}\right. } \end{aligned}$$

The family $(\partial _i(\eta _r)_j)_{r\geqslant 0}$ then locally uniformly converges to the continuous function $\alpha _{ij}$ on U, and it follows that the canonical 1-form $\eta$ is of class ${\mathcal {C}}^1$ on U, and hence on $\partial \!K$.

We shall now show that $\eta$ is a contact 1-form. Since $(\eta _r)_{r\geqslant 0}$ locally converges to $\eta$ in ${\mathcal {C}}^1$ topology, it holds that

$$\begin{aligned} \forall p \in \partial \!K, \forall u,v \in T_p\partial \!K,\quad {{\,\textrm{d}\,}}\!\eta |_p(u,v) = \lim _{r\rightarrow +\infty } {{\,\textrm{d}\,}}\!\eta _r|_p (u,v). \end{aligned}$$

Let $r\geqslant 0$, $p\in \partial \!K$ and $u,v \in T_p \partial \!K$. Consider smooth local extensions of u and v that are still denoted the same way. Then

$$\begin{aligned} \begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(u,v)&= u \cdot \eta _r(v) - v\cdot \eta _r(u) - \eta _r([u,v]) \\&= e^{-r}(Y_u \cdot g(Y_v,J\!\partial _r) - Y_v\cdot g(Y_u,J\!\partial _r) - g([Y_u,Y_v],J\!\partial _r))\\&= e^{-r}g(\nabla _{Y_u}Y_v - \nabla _{Y_v}Y_u - [Y_u,Y_v], J\!\partial _r)\\&\quad + e^{-r}(g(Y_v,\nabla _{Y_u}J\!\partial _r) - g(Y_u,\nabla _{Y_v}J\!\partial _r))\\&= e^{-r}(g(Y_v,JSY_u) - g(Y_u,JSY_v)). \end{aligned} \end{aligned}$$

(5.2.5)

Let $\{J\!\partial _r,\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame on a cylinder $E({\mathbb {R}}_+\times U)$ with $p\in U$. Since $e^{-r}g(Y_v,JSY_u) \underset{r\rightarrow +\infty }{\longrightarrow } \frac{1}{2}\sum _{i,j=1}^{2n} \eta ^i(v)\eta ^j(u)g(E_i,J\!E_i)$, it follows that

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta (u,v) = \frac{1}{2}\sum _{i,j=1}^{2n} (\eta ^i(v)\eta ^j(u) - \eta ^i(u)\eta ^j(v))g(E_i,J\!E_j). \end{aligned}$$

Setting $\omega _{ij} = g(E_i,J\!E_j)= g(e_i,Je_j)$, which are constants, the latter expression reads

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta = -\frac{1}{2} \sum _{i,j=1}^{2n} \omega _{ij}\,\eta ^i\wedge \eta ^j. \end{aligned}$$

(5.2.6)

Assume furthermore that the local orthonormal frame $\{\nu ,e_1,\ldots ,e_{2n}\}$ on U is chosen such that we have $Je_{2k-1}=e_{2k}$ for $k\in \{1,\ldots ,n\}$. In that case, the constants $\omega _{ij}$ are given by

$$\begin{aligned} \omega _{ij} = {\left\{ \begin{array}{ll} -1 &{} \text {if } i=2k-1, j = 2k,\\ 1 &{} \text {if } i= 2k, j= 2k-1,\\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

Equation (5.2.6) then yields the equality

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta = \sum _{k=1}^{n} \eta ^{2k-1}\wedge \eta ^{2k}. \end{aligned}$$

(5.2.7)

From this last expression we derive the equality $({{\,\textrm{d}\,}}\!\eta )^n =(n!)\, \eta ^1\wedge \cdots \wedge \eta ^{2n}$, and hence the equality $\eta \wedge ({{\,\textrm{d}\,}}\!\eta )^n =(n!)\, \eta \wedge \eta ^1 \wedge \cdots \wedge \eta ^{2n}$. Since $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ is a local coframe, $\eta \wedge ({{\,\textrm{d}\,}}\!\eta )^n$ is a volume form on U. It follows that $\eta$ is a contact form. Since ${{\,\textrm{d}\,}}\!\eta$ is a linear combination of wedge products of the local 1-forms $(\eta ^i)$, it follows by the very definition of $\xi$ that $\eta (\xi ) = 1$ and ${{\,\textrm{d}\,}}\!\eta (\xi ,\cdot ) =0$, hence $\xi$ is the Reeb vector field of $\eta$. The proof is now complete. $\square$

5.3 Regularity of the Carnot–Carathéodory metric

We shall now study the regularity of the Carnot–Carathéodory metric $\gamma _H$. To do so, we show that the local differential forms $\eta ^1,\ldots , \eta ^{2n}$, defined using a radially parallel orthonormal frame, have such regularity. The condition $\min \{a,b\}>1$ is too weak to ensure that it is of class ${\mathcal {C}}^1$, since the renormalization of $\eta ^j$ is of order $e^{-\frac{r}{2}}$ while that of $\eta$ is of order $e^{-r}$. We shall now show that the condition $\min \{a,b\}>\frac{3}{2}$ is sufficient. The proof is very similar to that of Theorem B. We first give a bound on the growth of $\nabla _{Y_u}E_j$.

Lemma 5.8

Let (M, g, J) be an (ALCH) manifold of order $a>\frac{1}{2}$ with an essential subset K. Let X be a radially parallel vector field on $\overline{M\setminus K}$, that is such that $\nabla _{\partial _r}X = 0$ and assume that $\Vert X\Vert _g=1$. Then there exists a constant $c>0$ such that

$$\begin{aligned} \Vert \nabla _{Y_u}X\Vert _g \leqslant c \Vert u\Vert _ge^r. \end{aligned}$$

Proof

Let $L = \Vert \nabla _{Y_u}X\Vert _g$, which is locally Lipschitz and hence almost everywhere differentiable. Since $\nabla _{\partial _r}X=0$, it holds that $\nabla _{\partial _r}(\nabla _{Y_u}X) = -R(\partial _r,Y_u)X$, and therefore, applying $\partial _r$ to $\frac{1}{2}L^2$ gives the almost everywhere inequality

$$\begin{aligned} L'L = -R(\partial _r,Y_u,X,\nabla _{Y_u}X) \quad a.e. \end{aligned}$$

Recall that $\Vert R\Vert _g$ is uniformly bounded (Lemma 3.3), and that since $a>\frac{1}{2}$, there exists $c_1>0$ such that $\Vert Y_v\Vert _g\leqslant c_1\Vert u\Vert _ge^r$ (Lemma 4.12). Hence, there exists $c>0$ such that

$$\begin{aligned} L'L \leqslant c\Vert u\Vert _ge^r\Vert X\Vert _g L \quad a.e. \end{aligned}$$

Since L is non-positive and $\Vert X\Vert _g=1$, it follows that

$$\begin{aligned} L' \leqslant c\Vert u\Vert _ge^r \quad a.e, \end{aligned}$$

which is true even at points where L vanishes since at those points, it achieves a minimum and its derivative thus vanishes. The result follows from a straightforward integration. $\square$

In particular, this Lemma applies to the vectors of a radially parallel orthonormal frame.

Proposition 5.9

Let (M, g, J) be an (ALCH) and (ALS) manifold of order a and b, with $\min \{a,b\}>\frac{3}{2}$, with an essential subset K. Let $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame on a cylinder $E({\mathbb {R}}_+\times U)$. Let u and v be local vector fields on $\partial \!K$. For all $j\in \{1,\ldots ,2n\}$, define

$$\begin{aligned} f^j(r,p) = g(\nabla _{Y_v}Y_u,E_j) + g(\nabla _{Y_u}E_j,Y_v). \end{aligned}$$

Then there exists a constant $c>0$ independent of (r, p, u, v) and a continuous function $\alpha ^j(p)$ on U such that

$$\begin{aligned} |f^j(r,p)-{e^\frac{r}{2}}\alpha ^j(p) |\leqslant c(\Vert u\Vert _g\Vert v\Vert _g + \Vert \nabla _{Y_v}Y_u\Vert _g){\left\{ \begin{array}{ll} e^{(2-\min \{a,b\})r} &{} \text {if } \min \{a,b\} < \frac{5}{2},\\ (r+1)e^{-\frac{r}{2}} &{} \text {if } \min \{a,b\} = \frac{5}{2},\\ e^{-\frac{r}{2}} &{} \text {if } \min \{a,b\}>\frac{5}{2}. \end{array}\right. } \end{aligned}$$

Proof

The proof is a strict adaptation of that of Proposition 5.7. The exact same computations show that

$$\begin{aligned} \begin{aligned} \partial _r\partial _rf^j&= (\nabla _{\partial _r}R)(\partial _r,E_j,Y_v,Y_u) + R(\partial _r,E_j,SY_v,Y_u) \\&\quad + R(\partial _r,E_j,Y_v,SY_u) -R(\partial _r,Y_v,SY_u,E_j)\\&\quad - (\nabla _{Y_v}R)(\partial _r,Y_u,\partial _r,E_j) - R(SY_v,Y_u,\partial _r,E_j)\\&\quad -R(\partial _r,\nabla _{Y_v}Y_u,\partial _r,E_j) -R(\partial _r,Y_u,SY_v,E_j) \\&\quad -R(\partial _r,Y_u,E_j,SY_v) -R(\partial _r,Y_v,\partial _r,\nabla _{Y_u}E_j), \end{aligned} \end{aligned}$$

and the exact same cancellations due to the symmetries of the Riemann tensor yield

$$\begin{aligned} \begin{aligned} \partial _r\partial _rf^j&= (\nabla _{\partial _r}R) (\partial _r,E_j,Y_v,Y_u) - (\nabla _{Y_v} R) (\partial _r,Y_u,\partial _r,E_j)\\&\quad + R(\partial _r,E_j,Y_v,SY_u) - R(\partial _r,Y_v,SY_u,E_j)\\&\quad -R(\partial _r,\nabla _{Y_v}Y_u,\partial _r,E_j) - R(\partial _r,Y_v,\partial _r,\nabla _{Y_u}E_j). \end{aligned} \end{aligned}$$

(5.3.1)

Define ${\textbf{k}}^j$ as

$$\begin{aligned} \begin{aligned} {\textbf{k}}^j&= R^0(\partial _r,E_j,Y_v,SY_u) - R^0(\partial _r,Y_v,SY_u,E_j)\\&\quad -R^0(\partial _r,\nabla _{Y_v}Y_u,\partial _r,E_j) - R^0(\partial _r,Y_v,\partial _r,\nabla _{Y_u}E_j), \end{aligned} \end{aligned}$$

so that Eq. (5.3.1) becomes

$$\begin{aligned} \begin{aligned} \partial _r\partial _rf^j - {\textbf{k}}^j&= (\nabla _{\partial _r}R) (\partial _r,E_j,Y_v,Y_u) - (\nabla _{Y_v} R) (\partial _r,Y_u,\partial _r,E_j)\\&\quad + (R-R^0)(\partial _r,E_j,Y_v,SY_u) - (R-R^0)(\partial _r,Y_v,SY_u,E_j)\\&\quad -(R-R^0)(\partial _r,\nabla _{Y_v}Y_u,\partial _r,E_j)\\&\quad - (R-R^0)(\partial _r,Y_v,\partial _r,\nabla _{Y_u}E_j). \end{aligned} \end{aligned}$$

(5.3.2)

From the computations of Lemma A.2 in the Appendix, it turns out that

$$\begin{aligned} \begin{aligned} {\textbf{k}}^j&= \frac{1}{4} g(SY_u,J\!\partial _r)g(Y_v,J\!E_j) -\frac{1}{4} g(SY_u,J\!E_j)g(Y_v,J\!\partial _r)\\&\quad -\frac{1}{4}g(SY_u,J\!\partial _r)g(Y_v,J\!E_j) - \frac{1}{2}g(SY_u,J\!E_j)g(Y_v,J\!\partial _r) \\&\quad +\frac{1}{4}g(\nabla _{Y_v}Y_u,E_j) \\&\quad +\frac{1}{4}g(Y_v,\nabla _{Y_u}E_j) + \frac{3}{4}g(SY_u,J\!E_j)g(Y_v,J\!\partial _r)\\&=\frac{1}{4}g(\nabla _{Y_v}Y_u,E_j) + \frac{1}{4}g(Y_v,\nabla _{Y_u}E_j)\\&= \frac{1}{4}f^j. \end{aligned} \end{aligned}$$

Thus, $f^j$ is a solution to the second order linear ODE

$$\begin{aligned} \partial _r\partial _rf^j - \frac{1}{4}f^j = h^j, \end{aligned}$$

with $h^j$ being equal to the right-hand side of (5.3.2). From classical second order linear ODE considerations, $f^j$ is given by

$$\begin{aligned} \begin{aligned} f^j(r,p)&= e^{\frac{r}{2}}\left( \frac{1}{2}f^j(0,p) + \partial _rf^j(0,p) + \int _0^r e^{-\frac{s}{2}}h^j(s,p){{\,\textrm{d}\,}}\!s\right) \\&\quad + e^{-\frac{r}{2}}\left( \frac{1}{2}f^j(0,p) - \partial _rf^j(0,p) - \int _0^r e^{\frac{s}{2}}h^j(s,p){{\,\textrm{d}\,}}\!s\right) . \end{aligned} \end{aligned}$$

The following upper bound is straightforward

$$\begin{aligned}\begin{aligned} |h^j|&\leqslant 2(\Vert R-R^0\Vert _g\Vert S\Vert _g + \Vert \nabla R\Vert _g)\Vert Y_u\Vert _g\Vert Y_v\Vert _g + \Vert R-R^0\Vert _g\Vert \nabla _{Y_v}Y_u\Vert _g \\&\quad + \Vert R-R^0\Vert _g\Vert Y_v\Vert _g\Vert \nabla _{Y_u}E_j\Vert _g. \end{aligned} \end{aligned}$$

Lemma 5.8 yields the existence of $c>0$ such that $\Vert \nabla _{Y_u}E_j\Vert _g\leqslant c\Vert u\Vert _ge^r$. It now follows from the (ALCH) and (ALS) conditions together with Lemmas 4.12 and 5.5 that there exists a constant $C>0$ such that

$$\begin{aligned} |h^j| \leqslant C(\Vert u\Vert _g\Vert v\Vert _g + \Vert (\nabla _{Y_v}Y_u)(p)\Vert _g)e^{(2-\min \{a,b\})r}, \end{aligned}$$

and therefore, we have the following upper bounds

$$\begin{aligned} \begin{aligned} |e^{-\frac{s}{2}}h^j(s,p)|&\leqslant C(\Vert u\Vert _g\Vert v\Vert _g + \Vert (\nabla _{Y_v}Y_u)(p)\Vert _g)e^{(\frac{3}{2}-\min \{a,b\})r}, \\ |e^{\frac{s}{2}}h^j(s,p)|&\leqslant C(\Vert u\Vert _g\Vert v\Vert _g + \Vert (\nabla _{Y_v}Y_u)(p)\Vert _g)e^{(\frac{5}{2}-\min \{a,b\})r}. \end{aligned} \end{aligned}$$

Let $\alpha ^j(p) = \frac{1}{2}f^j(0,p) + \partial _rf^j(0,p) + \int _0^{+\infty } e^{-\frac{s}{2}}h(s,p) {{\,\textrm{d}\,}}\!s$, which is continuous by the dominated convergence Theorem. The result follows from a strictly similar study than that of the proof of Proposition 5.7. $\square$

We are now able to prove our third main result, which we restate for the reader’s convenience.

Theorem C

Let (M, g, J) be an (ALCH) and (ALS) manifold of order a and b, with $\min \{a,b\}>\frac{3}{2}$, with an essential subset K such that the sectional curvature of $\overline{M\setminus K}$ is negative. Then the canonical Carnot-Carathéodory metric $\gamma _H$ has ${\mathcal {C}}^1$ regularity.

Proof

Let $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ be the local coframe associated to a radially parallel orthonormal frame $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ on a cylinder $E({\mathbb {R}}_+\times U)$. Let $k\in \{1,\ldots ,2n\}$ be fixed and $(\eta ^k_r)_{r\geqslant 0}$ be the local family of 1-forms that locally uniformly converges to $\eta ^k$ in ${\mathcal {C}}^0$ topology. Let $\{x^1,\ldots ,x^{2n+1}\}$ be local coordinates on an open subset of U, and write $\eta _r^k = \sum _{j=1}^{2n+1} (\eta _r^k)_j {{\,\textrm{d}\,}}\!x^j$. The partial derivatives of $\eta _r^k$ in these coordinates are given by

$$\begin{aligned} \partial _i (\eta ^k_r)(\partial _j) = e^{-\frac{r}{2}}\big ( g(\nabla _{Y_j}Y_i,E_k) + g(\nabla _{Y_i}E_k,Y_j)\big ), \end{aligned}$$

and it follows from Proposition 5.9 that they locally uniformly converges. Hence, $\eta ^k$ is of class ${\mathcal {C}}^1$.

The local coframe $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ has then been shown to be of class ${\mathcal {C}}^1$. Since the Carnot-Carathéodory metric is locally given by $\gamma _H = \sum _{i=1}^{2n}\eta ^i\otimes \eta ^i$ it follows that it has ${\mathcal {C}}^1$ regularity. This concludes the proof. $\square$

Remark 5.10

It is worth noting that in that case, although $\eta$ is not of class ${\mathcal {C}}^2$, its exterior differential ${{\,\textrm{d}\,}}\!\eta$ is of class ${\mathcal {C}}^1$, since it is locally expressed as a combination of $\{\eta ^i\wedge \eta ^j\}_{i,j\in \{1,\ldots ,2n\}}$.

6 The almost complex structure

6.1 Notations

Throughout this section, the Kähler manifold (M, g, J) is assumed to satisfy the (ALCH) and (ALS) conditions of orders $a,b>\frac{3}{2}$, with an essential subset K such that the sectional curvature of $\overline{M\setminus K}$ is negative. From Theorem C, $\partial \!K$ is endowed with a contact form $\eta$ of class ${\mathcal {C}}^1$ with contact structure $H = \ker \eta$, and with a ${\mathcal {C}}^1$ Carnot-Carathéodory metric $\gamma _H$ which is positive definite on H. Let $g_0 = g|_{\partial \!K}$ be the induced metric on $\partial \!K$, and recall that if $g_r = E(r,\cdot )^*g|_{\{\partial _r\}^{\perp }}$, Corollary 4.16 yields the existence of $c_0>0$ such that $g_0 \leqslant c_0e^{-r}g_r$.

For $r \geqslant 0$, we set $S_r = E(r,\cdot )^* S$ the pull-back of the shape operator thought as a field of endomorphisms of $\{\partial _r\}^{\perp }$. To any radially parallel orthonormal frame $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ is associated a local coframe $\{\eta _r,\eta ^1_r,\ldots ,\eta ^{2n}_r\}$ which locally converges in ${\mathcal {C}}^1$ topology to the local coframe $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$. Moreover, there exists a constant $c>0$ independent of the radially parallel orthonormal frame and from r, such that the differential forms $\{\eta _r,\eta ^1_r,\ldots ,\eta _r^{2n}\}$ and $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ all have $g_0$-norm less than c. By Proposition 4.8, it holds that

$$\begin{aligned} \max \{ e^r\Vert \eta _r-\eta \Vert _{g_0}, e^{\frac{r}{2}}\Vert \eta ^1_r-\eta ^1\Vert _{g_0},\ldots ,e^{\frac{r}{2}}\Vert \eta ^{2n}_r-\eta ^{2n}\Vert _{g_0}\} \leqslant c e^{-\frac{r}{2}}. \end{aligned}$$

(6.1.1)

If v is a vector field tangent to $\partial \!K$ and if $Z_v$ and $Z'_v$ are the vector fields asymptotic to $Y_v$ and $SY_v$, then Proposition 4.11 states that there exists $C>0$ such that

$$\begin{aligned} \max \{\Vert Y_v-Z_v\Vert _g,\Vert SY_v-Z'_v\Vert _g\} \leqslant C \Vert v\Vert _g e^{-\frac{r}{2}}. \end{aligned}$$

(6.1.2)

The dual frames of the local coframes $\{\eta _r,\eta ^1_r,\ldots ,\eta _r^{2n}\}$ and $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ are denoted by $\{\xi ^r,\xi ^r,\ldots ,\xi ^r_{2n}\}$ and $\{\xi ,\xi _1,\ldots ,\xi _{2n}\}$. Remark that $\xi ^r = E(r,\cdot )^* \left( e^rJ\!\partial _r\right)$ and $\xi ^r_j = E(r,\cdot )^* \left( e^{\frac{r}{2}}E_j\right)$. Since $\{\eta _r,\eta ^1_r,\ldots ,\eta _r^{2n}\}$ locally converges in ${\mathcal {C}}^1$ topology, so does $\{\xi ^r,\xi ^r,\ldots ,\xi ^r_{2n}\}$. It follows that $\{\xi ,\xi _1,\ldots ,\xi _{2n}\}$ is of class ${\mathcal {C}}^1$.

Let us define a field of endomorphisms on $\partial \!K$ related to the ambient almost complex structure J. Since J does not preserve $\{\partial _r\}^{\perp }$, it is not possible to pull it back on $\partial \!K$ by $E(r,\cdot )$. However, the tensor

$$\begin{aligned} \Phi = J - g(\cdot ,\partial _r)\otimes J\!\partial _r+ g(\cdot ,J\!\partial _r) \otimes \partial _r, \end{aligned}$$

stabilizes the distribution $\{\partial _r\}^{\perp }$.

Definition 6.1

For $r\geqslant 0$, let $\phi _r$ be defined by $\phi _r = E(r,\cdot )^* \Phi$.

Since $\Phi$ satisfies the two equalities $\Phi ^2 = -{{\,\textrm{Id}\,}}+ g(\cdot ,\partial _r)\otimes \partial _r+ g(\cdot , J\!\partial _r)\otimes J\!\partial _r$ and $\Phi ^3 = - \Phi$, we have the immediate Lemma.

Lemma 6.2

For all $r \geqslant 0$, $\phi _r^2 = -{{\,\textrm{Id}\,}}+ \eta _r\otimes \xi ^r$, and $\phi _r^3 = -\phi _r$.

Remark 6.3

Lemma 6.2 states that $(\partial \!K,\phi _r,\eta _r,\xi ^r)$ is an almost contact manifold in the sense of [8, 26, 27]. The following estimates are worth noting.

Lemma 6.4

There exists $c_1,c_2>0$ such that for all $r \geqslant 0$, it holds

• 1. $\Vert \phi _r\Vert _{g_0} \leqslant 1$,

• 2. $\Vert \phi _r \xi \Vert _{g_0} \leqslant c_1 e^{-r}$,

• 3. $\Vert \eta \circ \phi _r\Vert _{g_0} \leqslant c_2e^{-r}$.

Proof

• 1. Since $\Phi = \pi ^{\perp }\circ J \circ \pi ^{\perp }$, where $\pi ^{\perp }$ is the orthogonal projection on ${\partial _r}^{\perp }$ and J is an isometry, $\Phi$ has an operator norm less than or equal to 1. The result follows.

• 2. Since $\phi _r \xi ^r =0$, it holds that

  $$\begin{aligned} \begin{aligned} \Vert \phi _r\xi \Vert _{g_0}&= \Vert \phi _r(\xi - \xi ^r)\Vert _{g_0} \\&\leqslant \Vert \phi _r\Vert _g\Vert \xi -\xi ^r\Vert _{g_0} \\&\leqslant c_0e^{-\frac{r}{2}}\Vert Y_{\xi } - e^rJ\!\partial _r\Vert _{g_0} \\&\leqslant c_0C\Vert \xi \Vert _{g_0} e^{-r}, \end{aligned} \end{aligned}$$

  the inequalities being derived from the first point and from Eq. (6.1.2). To conclude, define $c_1 = c_0C \sup _{\partial \!K}\Vert \xi \Vert _g <+\infty$.

• 3. Since $\eta _r \circ \phi _r = 0$, it holds that

  $$\begin{aligned} \Vert \eta \circ \phi _r\Vert _{g_0} = \Vert (\eta -\eta _r)\circ \phi _r\Vert _{g_0} \leqslant \Vert \eta -\eta _r\Vert _{g_0} \Vert \phi _r\Vert _{g_0} \leqslant ce^{-\frac{3}{2}r}, \end{aligned}$$

  the latter inequality being derived from Eq. (6.1.1) and from the first point. The result is then true with $c_2=c$.

$\square$

6.2 Shape operator estimates

We shall now give estimates on the asymptotic behaviour of $(S_r)_{r\geqslant 0}$. We first prove that $S_r$ is asymptotic to the tensor $\frac{1}{2}({{\,\textrm{Id}\,}}+\eta \otimes \xi )$.

Lemma 6.5

There exists a constant $c'>0$ such that

$$\begin{aligned} \forall r \geqslant 0,\quad \Vert S_r - \frac{1}{2}({{\,\textrm{Id}\,}}+ \eta \otimes \xi ) \Vert _{g_0} \leqslant c' e^{-r}. \end{aligned}$$

Proof

Fix $v \in T\partial \!K$. From Corollary 4.16, there exists $c_0>0$ such that

$$\begin{aligned} \Vert S_rv - \frac{1}{2}(v + \eta (v)\xi )\Vert _{g_0}\leqslant c_0 e^{-\frac{r}{2}}\Vert SY_v - \frac{1}{2}(Y_v + \eta (v) Y_{\xi })\Vert _{g_0}. \end{aligned}$$

(6.2.1)

The local expressions of $Z_v$ and $Z_v'$ in a radially parallel orthonormal frame show that $\frac{1}{2}\eta (v)e^rJ\!\partial _r= Z_v' - \frac{1}{2}Z_v$. Therefore, it holds that

$$\begin{aligned} SY_v - \frac{1}{2}(Y_v+ \eta (v)e^rJ\!\partial _r) = SY_v - Z_v' - \frac{1}{2}(Y_v - Z_v) + \frac{1}{2}\eta (v)(e^rJ\!\partial _r- Y_{\xi }). \end{aligned}$$

By definition, $\eta (\xi ) =1$, and thus $e^rJ\!\partial _r= Z_{\xi }$. It follows from the triangle inequality that

$$\begin{aligned} \begin{aligned} \Vert SY_v - \frac{1}{2}(Y_v+ \eta (v)e^rJ\!\partial _r)\Vert _g&\leqslant \Vert SY_v-Z_v'\Vert _g + \frac{1}{2}\Vert Y_v-Z_v\Vert _g \\ {}&\quad + \frac{1}{2} |\eta (v)| \Vert Y_{\xi }-Z_{\xi }\Vert _g. \end{aligned} \end{aligned}$$

It now follows from the uniform bound on $\Vert \eta \Vert _{g_0}$ and from Eq. (6.1.2) that there exists $C>0$ such that

$$\begin{aligned} \Vert SY_v - \frac{1}{2}(Y_v+ \eta (v)e^rJ\!\partial _r)\Vert _g \leqslant C\Vert v\Vert _g(1+\Vert \xi \Vert _g)e^{-\frac{r}{2}}. \end{aligned}$$

Finally, Eq. (6.2.1) yields

$$\begin{aligned} \Vert S_rv - \frac{1}{2}(v + \eta (v)\xi )\Vert _{g_0}\leqslant c_0C(1+\Vert \xi \Vert _{g_0}) \Vert v\Vert _{g} e^{-r}. \end{aligned}$$

The result follows by setting $c' = c_0C(1+\sup _{\partial \!K}\Vert \xi \Vert _{g_0})$, which is finite since $\xi$ is a global vector field and $\partial \!K$ is compact. $\square$

Remark 6.6

In the model setting, the shape operator of concentric spheres is of the form $S = \coth r {{\,\textrm{Id}\,}}_{{\mathbb {R}}J\!\partial _r} + \frac{1}{2}\coth (\frac{r}{2}){{\,\textrm{Id}\,}}_{\{\partial _r,J\!\partial _r\}^{\perp }}$.

We shall now show that $S_r$ and $\phi _r$ asymptotically commute.

Lemma 6.7

There exists $c''>0$ such that

$$\begin{aligned} \forall r \geqslant 0,\quad \Vert S_r\phi _r - \phi _r S_r \Vert _{g_0} \leqslant c'' e^{-r}. \end{aligned}$$

Proof

First, write

$$\begin{aligned} \begin{aligned} S_r\phi _r&= \bigg (S_r - \frac{1}{2}({{\,\textrm{Id}\,}}+ \eta \otimes \xi )\bigg ) \phi _r + \frac{1}{2}\phi _r + \frac{1}{2} (\eta \circ \phi _r) \otimes \xi ,\\ \phi _r S_r&= \phi _r\bigg (S_r - \frac{1}{2}({{\,\textrm{Id}\,}}+ \eta \otimes \xi )\bigg ) + \frac{1}{2}\phi _r + \frac{1}{2} \eta \otimes (\phi _r \xi ). \end{aligned} \end{aligned}$$

It now follows from the triangle inequality that

$$\begin{aligned} \Vert S_r\phi _r - S_r\phi _r \Vert _{g_0} \leqslant 2\Vert \phi _r\Vert _{g_0} \Vert S_r - \frac{1}{2}({{\,\textrm{Id}\,}}+ \eta \otimes \xi )\Vert _{g_0} + \Vert \eta \circ \phi _r\Vert _{g_0} \Vert \xi \Vert _{g_0} + \Vert \eta \Vert _{g_0} \Vert \phi _r \xi \Vert _{g_0}. \end{aligned}$$

The result follows from Lemmas 6.4 and 6.5. $\square$

6.3 Convergence

We now show that $(\phi _r)_{r\geqslant 0}$ converges in ${\mathcal {C}}^1$ topology to some tensor $\phi$, and that the restriction of this tensor to H is an almost complex structure.

Proposition 6.8

The family $(\phi _r)_{r\geqslant 0}$ converges in ${\mathcal {C}}^1$ topology to a ${\mathcal {C}}^1$ tensor $\phi$ satisfying $\phi ^2 = -{{\,\textrm{Id}\,}}+ \eta \otimes \xi$, $\phi ^3 = -\phi$, $\eta \circ \phi = 0$ and $\eta \xi = 0$. In particular, $\phi$ preserves $H = \ker \eta$ and $({\phi |_H})^2 = - {{\,\textrm{Id}\,}}_H$.

Proof

Let $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$ be a radially parallel orthonormal frame on a cylinder $E({\mathbb {R}}_+\times U)$. By the very definition of $\Phi$, it holds that

$$\begin{aligned} \Phi = \sum _{j=1}^{2n} g(\cdot ,E_j)\otimes J\!E_j. \end{aligned}$$

Assume that the radially parallel orthonormal frame is a J-frame, that is for any $k\in \{1,\ldots ,n\}$, it holds that $J\!E_{2k-1} = J\!E_{2k}$. Then $\Phi$ has the nice expression

$$\begin{aligned} \Phi = \sum _{k=1}^n g(\cdot ,E_{2k-1})\otimes E_{2k} - g(\cdot ,E_{2k})\otimes E_{2k-1}, \end{aligned}$$

from which we deduce the following expression for $\phi _r$

$$\begin{aligned} \forall r \geqslant 0,\quad \phi _r = \sum _{k=1}^n \eta ^{2k-1}_r \otimes \xi ^r_{2k} - \eta ^{2k}_r \otimes \xi ^r_{2k-1}. \end{aligned}$$

From the convergence of the local coframe $\{\eta _r,\eta ^1_r,\ldots ,\eta _r^{2n}\}$ to $\{\eta ,\eta ^1,\ldots ,\eta ^{2n}\}$ and of the local frame $\{\xi ^r,\xi _1^r,\ldots ,\xi _{2n}^r\}$ to $\{\xi ,\xi _1,\ldots ,\xi _{2n}\}$ in ${\mathcal {C}}^1$ topology, it follows that on U, $(\phi _r)_{r\geqslant 0}$ converges in ${\mathcal {C}}^1$ topology to

$$\begin{aligned} \phi = \sum _{k=1}^n \eta ^{2k-1}\otimes \xi _{2k} - \eta ^{2k}\otimes \xi _{2k-1}. \end{aligned}$$

Notice that this does not depend on the chosen radially parallel J-orthonormal frame, and $\phi _r\rightarrow \phi$ in ${\mathcal {C}}^1$ topology on $\partial \!K$. Taking the limit as $r\rightarrow +\infty$ in Lemmas 6.2 and 6.4 concludes the proof. $\square$

Definition 6.9

The almost complex structure $J_H$ on H is defined as the restriction of $\phi$ to H.

6.4 Integrability

Since $J_H$ is an almost complex structure, the complexified bundle $H\otimes {\mathbb {C}}$ splits as $H\otimes {\mathbb {C}}= H^{1,0} \oplus H^{0,1}$, where $H^{1,0} = \ker \{J_H - i {{\,\textrm{Id}\,}}\}= \{X+iJ_HX \mid X\in H\}$ and $H^{0,1} = \ker \{ J_H + i {{\,\textrm{Id}\,}}\}$. Since H and $J_H$ are both of class ${\mathcal {C}}^1$, the Lie bracket of sections of $H^{1,0}$ makes sense. It is then possible to ask whether $J_H$ is integrable, that is, if $H^{1,0}$ is stable under the Lie bracket. However, $J_H$ is defined as the restriction of the limit of a family of tensors $(\phi _r)$, which does not preserve H, and it is not clear what condition on $\phi _r|_{\ker \eta _r}$ would ensure that $J_H$ is integrable. Considering the whole tensor $\phi$ is thus more convenient, as the following study shows.

Recall that $\phi$ satisfies $\phi ^3 = -\phi$. The complexified tangent bundle $T\partial \!K\otimes {\mathbb {C}}$ then splits into the direct sum of the eigenspaces of $\phi$ as

$$\begin{aligned} T\partial \!K\otimes {\mathbb {C}}= \ker \phi \oplus \ker \{\phi - i{{\,\textrm{Id}\,}}\} \oplus \ker \{\phi + i {{\,\textrm{Id}\,}}\} = {\mathbb {C}}\xi \oplus H^{1,0} \oplus H^{0,1}. \end{aligned}$$

We still denote by $\phi$ and $\eta$ the complex-linear extensions of $\phi$ and $\eta$. Any complex vector field $V \in \Gamma (T\partial \!K)\otimes {\mathbb {C}}$ reads $V = \eta (V)\xi + V^{1,0} + V^{0,1}$, with $\phi V^{1,0}=iV^{1,0}$ and $\phi V^{0,1}=-iV^{0,1}$. Since $\phi \xi = 0$, it follows that $V+i\phi V = \eta (V)\xi + 2 V^{0,1}$. Hence, a complex vector field V on $\partial \!K$ is a section of $H^{1,0}$ if and only if $V+i\phi V = 0$. Finally, it follows that $H^{1,0}$ is integrable if and only if

$$\begin{aligned} \forall u,v \in \Gamma (H), \quad [u-i\phi u,v-i\phi v] + i \phi [u-i\phi u, v-i\phi v] = 0. \end{aligned}$$

The Nijenhuis tensor $N_A$ of a field of endomorphisms A is defined by

$$\begin{aligned} \forall X,Y,\quad N_A(X,Y) = -A^2[X,Y] - [AX,AY] + A[AX,Y] + A[X,AY]. \end{aligned}$$

The integrability of $J_H$ is related to the Nijenhuis tensor of $\phi$ by the following Lemma.

Lemma 6.10

For any vector fields u and v on $\partial \!K$, setting $V = [u-i\phi u, v-i\phi v]$, it holds that

$$\begin{aligned} \begin{aligned} V+i\phi V&= N_{\phi }(u,v) + \eta ([u,v])\xi + i\phi N_{\phi }(u,v)- i (\eta ([\phi u,v]) + \eta ([u,\phi v])) \xi . \end{aligned} \end{aligned}$$

Proof

Extending the Lie bracket ${\mathbb {C}}$-linearly, it holds that

$$\begin{aligned} \begin{aligned} V +i\phi V&= [u,v] - [\phi u, \phi v] -i [\phi u,v] - i [u,\phi v]\\&\quad + i \phi \big ( [u,v] - [\phi u, \phi v] -i [\phi u,v] - i [u,\phi v]\big )\\&= [u,v] - [\phi u, \phi v] + \phi [\phi u, v] + \phi [ u,\phi v]\\&\quad + i \big ( \phi [u,v] - \phi [\phi u, \phi v] -[\phi u,v] - [u, \phi v] \big ). \end{aligned} \end{aligned}$$

The equalities $\phi ^2 = -{{\,\textrm{Id}\,}}+ \eta \otimes \xi$ and $\phi ^3 = -\phi$ now yield

$$\begin{aligned} \begin{aligned} V+i\phi V&= (-\phi ^2 + \eta \otimes \xi )[u,v] - [\phi u, \phi v] + \phi [\phi u, v] + \phi [ u,\phi v]\\&\quad + i \big ( -\phi ^3[u,v] - \phi [\phi u, \phi v] + (\phi ^2 - \eta \otimes \xi )[\phi u, v] \\&\quad + (\phi ^2 - \eta \otimes \xi )[u,\phi v] \big ) \\&= -\phi ^2[u,v] -[\phi u, \phi v] + \phi [\phi u, v] + \phi [u,\phi v] + \eta ([u,v])\xi \\&\quad + i \phi \big ( -\phi ^2[u,v] -[\phi u, \phi v] + \phi [\phi u, v] + \phi [u,\phi v]\big )\\&\quad - i\big ( \eta ([\phi u,v]) + \eta ([u,\phi v]) \big ) \xi . \end{aligned} \end{aligned}$$

The result follows from the definition of the Nijenhuis tensor $N_{\phi }$. $\square$

Proposition 6.11

The almost complex structure $J_H$ is integrable if and only if the two following conditions are satisfied:

• 1. $N_{\phi }|_{H\times H} = {{\,\textrm{d}\,}}\!\eta |_{H\times H} \otimes \xi$,

• 2. ${{\,\textrm{d}\,}}\!\eta |_{H\times H}(J_H\cdot ,\cdot ) = -{{\,\textrm{d}\,}}\!\eta |_{H\times H}(\cdot ,J_H \cdot )$.

Proof

Let u and v be tangent to the distribution H and $V=[u-i\phi u, v-i \phi v]$. Then, V is tangent to $H^{1,0}$ if and only if $V+ i \phi V = 0$. Identifying the real and imaginary parts, Lemma 6.10 then states that V is tangent to $H^{1,0}$ if and only if

$$\begin{aligned} {\left\{ \begin{array}{ll} 0&{}=N_{\phi }(u,v) + \eta ([u,v])\xi , \\ 0&{}=\phi N_{\phi }(u,v) - \big (\eta ([\phi u, v]) + \eta ([u,\phi v])\big ) \xi . \end{array}\right. } \end{aligned}$$

Since $\phi \xi = 0$, this latter system is equivalent to

$$\begin{aligned} {\left\{ \begin{array}{ll} N_{\phi }(u,v) &{}=- \eta ([u,v])\xi , \\ \eta ([\phi u, v]) &{}= - \eta ([u,\phi v]). \end{array}\right. } \end{aligned}$$

By definition of u and v, $\eta (u)=\eta (v) = 0$. Moreover, since $\eta \circ \phi = 0$, it follows that $\eta (\phi u) = \eta (\phi v) = 0$. Therefore, it holds that

$$\begin{aligned} {\left\{ \begin{array}{ll} {{\,\textrm{d}\,}}\!\eta ([u,v]) = u \cdot \eta (v) - v\cdot \eta (u) - \eta ([u,v] = -\eta ([u,v])),\\ {{\,\textrm{d}\,}}\!\eta ([\phi u, v]) = (\phi u) \cdot \eta (v) - v \cdot \eta (\phi u) - \eta ([\phi u,v]) = -\eta ([\phi u, v]),\\ {{\,\textrm{d}\,}}\!\eta ([ u,\phi v]) = u \cdot \eta (\phi v) - (\phi v) \cdot \eta ( u) - \eta ([u,\phi v]) = -\eta ([ u,\phi v]). \end{array}\right. } \end{aligned}$$

Finally, V is tangent to $H^{1,0}$ if and only if $N_{\phi }(u,v) = {{\,\textrm{d}\,}}\!\eta (u,v) \xi$ and ${{\,\textrm{d}\,}}\!\eta (\phi u, v) = - {{\,\textrm{d}\,}}\!\eta (u, \phi v)$. $\square$

Lemma 6.12

There exists ${\widetilde{c}}>0$ such that for all $r\geqslant 0$, $p\in \partial \!K$ and $u,v\in T_p\partial \!K$, we have

$$\begin{aligned} |{{\,\textrm{d}\,}}\!\eta _r(\phi _r u,v) + {{\,\textrm{d}\,}}\!\eta _r(u,\phi _r v)| \leqslant {\widetilde{c}}\Vert u\Vert _g\Vert v\Vert _g e^{-\frac{r}{2}}. \end{aligned}$$

Proof

Recall that Eq. (5.2.5) gives, for all $r \geqslant 0$ and u, v tangent to $\partial \!K$

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(u,v) = e^{-r} \big ( g(Y_v,JSY_u) - g(Y_u,JSY_v)\big ). \end{aligned}$$

Since J is skew-symmetric, it holds that

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(u,v) = e^{-r}\big ( g(SY_u,JY_v) - g(SY_v,JY_u)\big ). \end{aligned}$$

Notice that since $Y_u$ and $Y_v$ are normal to $\partial _r$, this latter expression reads

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(u,v) = e^{-r}\big ( g(SY_u,\Phi Y_v) - g(SY_v,\Phi Y_u)\big ). \end{aligned}$$

It thus holds that

$$\begin{aligned} \begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(\phi _ru,v)&= e^{-r}\big ( g(S\Phi Y_u,\Phi Y_v) - g(SY_v,\Phi ^2 Y_u)\big ),\\ {{\,\textrm{d}\,}}\!\eta _r(u,\phi _r v)&= e^{-r}\big ( g(SY_u,\Phi ^2 Y_v) - g(S\Phi Y_v,\Phi Y_u)\big ). \end{aligned} \end{aligned}$$

The symmetry of S now yields

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(\phi _ru,v) + {{\,\textrm{d}\,}}\!\eta _r(u,\phi _r v) = e^{-r} \big (g(SY_u,\Phi ^2Y_v) - g(SY_v,\Phi ^2 Y_u) \big ). \end{aligned}$$

Since $\Phi ^2Y_u = -Y_u + g(Y_u,J\!\partial _r)\otimes J\!\partial _r$, and similarly for $Y_v$, it follows that

$$\begin{aligned} \begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(\phi _ru,v) + {{\,\textrm{d}\,}}\!\eta _r(u,\phi _r v)&= e^{-r} \big (g(SY_v,J\!\partial _r)g(Y_u,J\!\partial _r) \\&\quad - g(SY_u,J\!\partial _r)g(Y_v,J\!\partial _r) \big ). \end{aligned} \end{aligned}$$

Using the symmetry of S and writing $SJ\!\partial _r= (SJ\!\partial _r- J\!\partial _r) + J\!\partial _r$ now shows that

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta _r(\phi _ru,v) + {{\,\textrm{d}\,}}\!\eta _r(u,\phi _r v) = \eta _r(u)g(Y_v,SJ\!\partial _r-J\!\partial _r) - \eta _r(v)g(Y_u,SJ\!\partial _r- J\!\partial _r). \end{aligned}$$

Recall that there exist constants $c_1,c_2>0$ such that $\Vert \eta _r\Vert _g \leqslant c_1$ and $\Vert Y_w\Vert _g \leqslant c_2\Vert w\Vert _ge^r$ for any w. Thus

$$\begin{aligned} |{{\,\textrm{d}\,}}\!\eta _r(\phi _ru,v) + {{\,\textrm{d}\,}}\!\eta _r(u,\phi _r v)| \leqslant c\Vert u\Vert _g\Vert v\Vert _g \Vert Se^rJ\!\partial _r- e^rJ\!\partial _r\Vert _g. \end{aligned}$$

To conclude, it suffices to control $\Vert Se^rJ\!\partial _r- e^rJ\!\partial _r\Vert _g$. Recall that $e^r J\!\partial _r= Z_{\xi } = Z'_{\xi }$. It now follows from the triangle inequality, the fact that S is uniformly bounded, and equation (6.1.2) that there exists $c_3 >0$ such that

$$\begin{aligned} \begin{aligned} \Vert Se^rJ\!\partial _r- e^rJ\!\partial _r\Vert _g&= \Vert S(Z_{\xi } - Y_{\xi }) + SY_{\xi } - Z_{\xi }'\Vert _g\\&\leqslant \Vert S\Vert _g\Vert Y_{\xi }-Z_{\xi }\Vert _g + \Vert SY_{\xi } - Z'_{\xi }\Vert _g\\&\leqslant c_3 \Vert \xi \Vert _ge^{-\frac{r}{2}}. \end{aligned} \end{aligned}$$

The proof follows by setting ${\widetilde{c}}=c_1c_2c_3 \sup _{\partial \!K}\Vert \xi \Vert _g < + \infty$ since $\xi$ is a continuous vector field on $\partial \!K$ compact. $\square$

Proposition 6.13

Let u and v be vector fields on $\partial \!K$. Then it holds that ${{\,\textrm{d}\,}}\!\eta (\phi u,v) = -{{\,\textrm{d}\,}}\!\eta (u,\phi v)$. In particular, ${{\,\textrm{d}\,}}\!\eta |_{H\times H}(J_H\cdot ,\cdot ) = -{{\,\textrm{d}\,}}\!\eta |_{H\times H}(\cdot ,J_H\cdot )$.

Proof

Recall that $(\phi _r)_{r\geqslant 0}$ and $({{\,\textrm{d}\,}}\!\eta _r)_{r\geqslant 0}$ locally uniformly converge to $\phi$ and ${{\,\textrm{d}\,}}\!\eta$. The result follows by taking the limit in Lemma 6.12 as $r\rightarrow +\infty$. $\square$

Proposition 6.13 shows that the condition 2. of Proposition 6.11 is satisfied. Let us show that the condition 1. is also satisfied.

Lemma 6.14

For all $r\geqslant 0$, $p\in \partial \!K$ and $u,v\in T_p\partial \!K$, we have

$$\begin{aligned} \begin{aligned} N_{\phi _r}(u,v)&= e^r\eta _r(v)(\phi _r S_r u - S_r\phi _r u) - e^r\eta _r(v)(\phi _r S_r v - S_r \phi _r v)\\&\quad + {{\,\textrm{d}\,}}\!\eta _r(u,v)\xi (r). \end{aligned} \end{aligned}$$

Proof

For X and Y vector fields on $\overline{M\setminus K}$, it holds that

$$\begin{aligned} \begin{aligned} -\Phi ^2[X,Y]&= -\Phi ^2 \nabla _XY + \Phi ^2 \nabla _YX,\\ -[\Phi X,\Phi Y]&= -\nabla _{\Phi X}(\Phi Y) + \nabla _{\Phi Y}(\Phi X)\\&= -(\nabla _{\Phi X}\Phi )Y - \Phi \nabla _{\Phi X}Y + (\nabla _{\Phi Y}\Phi )X + \Phi \nabla _{\Phi Y}X,\\ \Phi [\Phi X,Y]&= \Phi \nabla _{\Phi X} Y - \Phi \nabla _Y (\Phi X)\\&= \Phi \nabla _{\Phi X} Y - \Phi (\nabla _Y\Phi )X - \Phi ^2 \nabla _{X}Y, \text { and}\\ \Phi [X,\Phi Y]&= \Phi \nabla _X(\Phi Y) - \Phi \nabla _{\Phi Y} X\\&= \Phi (\nabla _X\Phi )Y + \Phi ^2\nabla _XY - \Phi \nabla _{\Phi Y}X. \end{aligned} \end{aligned}$$

Recall that $N_{\Phi }(X,Y) = -\Phi ^2[X,Y] -[\Phi X, \Phi Y] + \Phi [\Phi X, Y] + \Phi [X,\Phi Y]$. Therefore,

$$\begin{aligned} N_{\Phi }(X,Y) = \Phi (\nabla _{X}\Phi )Y - (\nabla _{\Phi X}\Phi )Y + (\nabla _{\Phi Y}\Phi )X - \Phi (\nabla _{Y}\Phi )X. \end{aligned}$$

Recall that $\nabla g = 0$, $\nabla J = 0$, $\nabla \partial _r= S$ and $\nabla J\!\partial _r= J S$. Hence, by the very definition of $\Phi$, it holds that

$$\begin{aligned} \begin{aligned} (\nabla _{X}\Phi ) Y&= -g(Y,SX) J\!\partial _r- g(Y,\partial _r)JSX\\&\quad + g(Y,JSX) \partial _r+ g(Y,J\!\partial _r) SX. \end{aligned} \end{aligned}$$

Since $\Phi \partial _r= \Phi J\!\partial _r= 0$, it holds that

$$\begin{aligned} \Phi (\nabla _X\Phi ) Y = -g(Y,\partial _r)\Phi JSX + g(Y,J\!\partial _r)\Phi SX. \end{aligned}$$

Similarly, the following equality holds

$$\begin{aligned} \begin{aligned} (\nabla _{\Phi X}\Phi )Y&= -g(Y,S\Phi X) J\!\partial _r- g(Y,\partial _r)JS\Phi X \\&\quad + g(Y,JS\Phi X)\partial _r+ g(Y,J\!\partial _r)S\Phi X. \end{aligned} \end{aligned}$$

It follows by the anti-symmetry in X and Y that

$$\begin{aligned} \begin{aligned} N_{\Phi }(X,Y)&= g(Y,J\!\partial _r)( \Phi SX - S\Phi X) - g(X,J\!\partial _r) (\Phi SY -S\Phi Y)\\&\quad +(g(Y,S\Phi X) - g(X,S\Phi Y)) J\!\partial _r\\&\quad + (g(Y,JS\Phi X) - g(X,JS\Phi Y)) \partial _r\\&\quad - g(Y,\partial _r) JS\Phi X + g(X,\partial _r) JS\Phi Y. \end{aligned} \end{aligned}$$

(6.4.1)

Notice that $g(X,JS\Phi Y) = -g(JX, S\Phi Y)$. Assume from now that X is orthogonal to $\partial _r$. Then JX is orthogonal to $J\!\partial _r$. Since $S\Phi Y$ is orthogonal to $\partial _r$, it follows that $g(JX, S\Phi Y) = g(\Phi X,S\Phi Y)$. In particular, if X and Y are both orthogonal to $\partial _r$, the terms in the third and fourth lines of Eq. (6.4.1) vanish. Hence, if u and v are vector fields on $\partial \!K$, it holds that

$$\begin{aligned} \begin{aligned} N_{\Phi }(Y_u,Y_v)&= g(Y_v,J\!\partial _r)(\Phi S Y_u - S\Phi Y_u) - g(Y_u,J\!\partial _r)(\Phi S Y_v - S\Phi Y_v)\\&\quad + (g(Y_v,S\Phi Y_u) - g(Y_u,S\Phi Y_v)) J\!\partial _r. \end{aligned} \end{aligned}$$

Recall from Eq. (5.2.5) that ${{\,\textrm{d}\,}}\!\eta _r(u,v) = e^{-r}(g(Y_v,JSY_u) - g(Y_u,JSY_v))$. In addition, notice that

$$\begin{aligned} g(Y_v,JSY_u) = -g(JY_v,SY_u) = -g(SJY_v,Y_u) = -g(S\Phi Y_v,Y_u). \end{aligned}$$

Hence, $g(Y_v,S\Phi Y_u) - g(Y_u,S\Phi Y_v) = e^r{{\,\textrm{d}\,}}\!\eta _r(u,v)$. The result now follows from the equality $E(r,\cdot )^* e^rJ\!\partial _r= \xi (r)$, the definition of $\eta _r$ and the properties of the pull-back. $\square$

Proposition 6.15

The restriction $N_{\phi }|_{H\times H}$ is equal to the tensor ${{\,\textrm{d}\,}}\!\eta \otimes \xi$.

Proof

By the convergence in ${\mathcal {C}}^1$ topology of $\phi _r$ to $\phi$, it follows that for all u, v, tangent to H, $N_{\phi _r}(u,v) \rightarrow N_{\phi }(u,v)$ as $r\rightarrow +\infty$. Since $H= \ker \eta$, it follows from Eq. (6.1.1) that there exists $c>0$ such that $|e^r\eta _r(u)| \leqslant c\Vert u\Vert _ge^{-\frac{r}{2}}$ and $|\eta _r(v)| \leqslant c\Vert v\Vert _ge^{-\frac{r}{2}}$. It then follows from Lemma 6.7 that there exists $c'>0$ such that

$$\begin{aligned} \forall r \geqslant 0, \forall u,v \in \Gamma (H),\quad \Vert N_{\phi _r}(u,v) - {{\,\textrm{d}\,}}\!\eta _r(u,v)\xi ^r\Vert _g \leqslant c' \Vert u\Vert _g\Vert v\Vert _{g_0} e^{-\frac{3r}{2}}. \end{aligned}$$

The result follows by taking the limit as $r\rightarrow +\infty$. $\square$

We shall now highlight the link between the canonical form $\eta$, the Carnot–Carathéodory metric $\gamma _H$, and the almost-complex structure $J_H$.

Proposition 6.16

The Carnot–Carathéodory metric $\gamma _H$, the exterior differential of the canonical 1-form at infinity $\eta$ and the tensor $\phi$ are related by the equality $\gamma _H = {{\,\textrm{d}\,}}\!\eta (\cdot ,\phi \cdot )$. In particular, the restriction of $\gamma _H$ to $H\times H$ is given by ${{\,\textrm{d}\,}}\!\eta |_{H\times H}(\cdot ,J_H,\cdot )$.

Proof

Fix a radially parallel J-orthonormal frame $\{\partial _r,J\!\partial _r,E_1,\ldots ,E_{2n}\}$. From equation (5.2.7), it holds that ${{\,\textrm{d}\,}}\!\eta$ locally reads

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta = \sum _{k=1}^n \eta ^{2k-1}\wedge \eta ^{2k} = \sum _{k=1}^{n} \eta ^{2k-1}\otimes \eta ^{2k} - \eta ^{2k}\otimes \eta ^{2k-1}, \end{aligned}$$

while $\gamma _H$ is locally expressed by

$$\begin{aligned} \gamma _H = \sum _{j=1}^{2n} \eta ^j\otimes \eta ^j. \end{aligned}$$

Since the radially parallel orthonormal frame is chosen to be a J-orthonormal frame, it holds that for all $r\geqslant 0$, and for all $k\in \{1,\ldots ,n\}$,

$$\begin{aligned} {\left\{ \begin{array}{ll} \eta _r^{2k-1}\circ \phi _r &{}= - \eta _r^{2k},\\ \eta _r^{2k}\circ \phi _r &{}= \eta _r^{2k-1}. \end{array}\right. } \end{aligned}$$

Taking the limit as $r\rightarrow +\infty$ shows that for all $k\in \{1,\ldots ,n\}$, $\eta ^{2k-1}\circ \phi = - \eta ^{2k}$ and $\eta ^{2k}\circ \phi =\eta ^{2k-1}$. It follows that

$$\begin{aligned} {{\,\textrm{d}\,}}\!\eta (\cdot ,\phi \cdot ) = \sum _{k=1}^n \eta ^{2k-1}\otimes \eta ^{2k-1}+\eta ^{2k}\otimes \eta ^{2k} = \gamma _H, \end{aligned}$$

which concludes the proof. $\square$

We are now able to prove the last of our main Theorems (we restate the hypotheses for the reader’s convenience).

Theorem D

Let (M, g, J) be a complete non-compact Kähler manifold with an essential subset K, such that the sectional curvature of $\overline{M\setminus K}$ is negative. Assume that M is (ALCH) and (ALS) of order a and b with $\min \{a,b\} >\frac{3}{2}$. Then $(\partial \!K,H,J_H)$ is a strictly pseudoconvex CR manifold of class ${\mathcal {C}}^1$.

Proof

Let $\eta$ be the canonical form, $\gamma _H$ be the Carnot–Carathéodory metric and $\phi$ be defined as in Definition 6.9. From Proposition 6.8, $\phi$ induces a ${\mathcal {C}}^1$ almost complex structure $J_H = \phi |_H$ on the distribution $H=\ker \eta$, which is contact (Theorem B). It follows from propositions 6.13 and 6.15 that $\phi$ satisfies

$$\begin{aligned} \forall u,v \in \Gamma (H),\quad {\left\{ \begin{array}{ll} N_{\phi }(u,v) &{}= {{\,\textrm{d}\,}}\!\eta (u,v) \xi , \text { and}\\ {{\,\textrm{d}\,}}\!\eta (\phi u, v) &{}= -{{\,\textrm{d}\,}}\!\eta (u,\phi v). \end{array}\right. } \end{aligned}$$

It then follows from Proposition 6.11 that $J_H$ is integrable, and $(\partial \!K, H,J_H)$ is a CR manifold of class ${\mathcal {C}}^1$. From Proposition 6.16, it holds that ${{\,\textrm{d}\,}}\!\eta |_{H\times H}(\cdot ,J_H \cdot ) =\gamma _H$, and since $\gamma _H$ is positive definite on H, it follows that $(\partial \!K,H,J_H)$ is strictly pseudoconvex. This concludes the proof. $\square$

Remark 6.17

If M has real dimension 4, then the contact structure on the boundary of M is of rank 2. Since almost complex structures (when they are at least of class ${\mathcal {C}}^1$) are always integrable in dimension 2, the proof of Theorem D is thus considerably reduced in that specific case.

Appendix A: Curvature computations

Lemma A.1

In the context of Proposition 5.7:

$$\begin{aligned} R^0(\partial _r,J\!\partial _r,Y_v,SY_u)= & {} -\frac{1}{2}g(SY_u,JY_v),\\ R^0(\partial _r,Y_v,SY_u,J\!\partial _r)= & {} \frac{1}{4}g(SY_u,JY_v),\\ R^0(\partial _r,\nabla _{Y_v}Y_u,\partial _r,J\!\partial _r)= & {} -g(\nabla _{Y_v}Y_u,J\!\partial _r), \text { and}\\ R^0(\partial _r,Y_v,\partial _r,\nabla _{Y_u}J\!\partial _r)= & {} -\frac{1}{4}g(Y_v,\nabla _{Y_u}J\!\partial _r). \end{aligned}$$

Proof

Since $\partial _r\perp SY_u$, $\partial _r\perp Y_v$, $J\!\partial _r\perp JSY_u$ and $J\!\partial _r\perp JY_v$, it follows that

$$\begin{aligned} \begin{aligned} R^0(\partial _r,J\!\partial _r,Y_v,SY_u)&= \frac{1}{4}\big ( g(\partial _r,SY_u)g(J\!\partial _r,Y_v) - g(\partial _r,Y_v)g(J\!\partial _r,SY_u)\\&\quad + g(\partial _r,JSY_u)g(J\!\partial _r,JY_v) - g(\partial _r,JY_v)g(J\!\partial _r,JSY_u)\\&\quad + 2 g(\partial _r,J^2\partial _r)g(SY_u,JY_v)\big )\\&= \frac{1}{4}(2g(\partial _r,-\partial _r)g(SY_u,JY_v))\\&= -\frac{1}{2}g(SY_u,JY_v). \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \begin{aligned} R^0(\partial _r,Y_v,SY_u,J\!\partial _r)&= \frac{1}{4}\big ( g(\partial _r,J\!\partial _r)g(Y_v,SY_u) - g(\partial _r,SY_u)g(Y_v,J\!\partial _r) \\&\quad + g(\partial _r,J^2\partial _r)g(Y_v,JSY_u) - g(\partial _r,JSY_u)g(Y_v,J^2\partial _r)\\&\quad +2g(\partial _r,JY_v)g(J\!\partial _r,SY_u)\big )\\&= \frac{1}{4}(-g(Y_v,JSY_u))\\&= \frac{1}{4}g(SY_u,JY_v). \end{aligned} \end{aligned}$$

In addition, since J is skew-symmetric,

$$\begin{aligned} \begin{aligned} R^0(\partial _r,\nabla _{Y_v}Y_u,\partial _r,J\!\partial _r)&= \frac{1}{4}\big ( g(\partial _r,J\!\partial _r)g(\nabla _{Y_v}Y_u,\partial _r) - g(\partial _r,\partial _r)g(\nabla _{Y_v}Y_u,J\!\partial _r)\\&\quad +g(\partial _r,J^2\partial _r)g(\nabla _{Y_v}Y_u,J\!\partial _r) - g(\partial _r,J\!\partial _r)g(\nabla _{Y_v}Y_u,J\!\partial _r) \\&\quad + 2g(\partial _r,J\nabla _{Y_v}Y_u)g(J\!\partial _r,J\!\partial _r)\big )\\&= \frac{1}{4}\big ( -g(\nabla _{Y_v}Y_u,J\!\partial _r) -g(\nabla _{Y_v}Y_u,J\!\partial _r) - 2g(\nabla _{Y_v}Y_u,J\!\partial _r)\big )\\&= -g(\nabla _{Y_v}Y_u,J\!\partial _r). \end{aligned} \end{aligned}$$

Finally, since J is skew-symmetric and parallel, it holds that

$$\begin{aligned} \begin{aligned} R^0(\partial _r,Y_v,\partial _r,\nabla _{Y_u}J\!\partial _r)&= \frac{1}{4}\big ( g(\partial _r,\nabla _{Y_u}J\!\partial _r)g(Y_v,\partial _r)-g(\partial _r,\partial _r) g(Y_v,\nabla _{Y_u}J\!\partial _r) \\&\quad + g(\partial _r,J\nabla _{Y_u}J\!\partial _r)g(Y_v,J\!\partial _r)-g(\partial _r,J\!\partial _r)g(Y_v,\nabla _{Y_u}J\!\partial _r)\\&\quad + 2g(\partial _r,JY_v)g(\nabla _{Y_u}J\!\partial _r,J\!\partial _r)\big )\\&= -\frac{1}{4}g(\nabla _{Y_u}J\!\partial _r,Y_v) + \frac{1}{2}g(\partial _r,JY_v)g(\nabla _{Y_u}J\!\partial _r,J\!\partial _r). \end{aligned} \end{aligned}$$

To conclude, note that since $\Vert J\!\partial _r\Vert _g^2 =1$ is constant, then $g(\nabla _{Y_u}J\!\partial _r,J\!\partial _r) = 0$. $\square$

Lemma A.2

In the context of Proposition 5.9:

$$\begin{aligned} R^0(\partial _r,E_j,Y_v,SY_u)= & {} \frac{1}{4} g(SY_u,J\!\partial _r)g(Y_v,J\!E_j) -\frac{1}{4} g(SY_u,J\!E_j)g(Y_v,J\!\partial _r),\\ R^0(\partial _r,Y_v,SY_u,E_j)= & {} \frac{1}{4}g(SY_u,J\!\partial _r)g(Y_v,J\!E_j) + \frac{1}{2}g(SY_u,J\!E_j)g(Y_v,J\!\partial _r),\\ R^0(\partial _r,\nabla _{Y_v}Y_u,\partial _r,E_j)= & {} -\frac{1}{4}g(\nabla _{Y_v}Y_u,E_j), \text { and}\\ R^0(\partial _r,Y_v,\partial _r,\nabla _{Y_u}E_j)= & {} -\frac{1}{4}g(Y_v,\nabla _{Y_u}E_j) - \frac{3}{4}g(SY_u,J\!E_j)g(Y_v,J\!\partial _r). \end{aligned}$$

Proof

Since $\partial _r\perp SY_u$, $\partial _r\perp Y_v$ and $\partial _r\perp J\!E_j$, it holds that

$$\begin{aligned} \begin{aligned} R^0(\partial _r,E_j,Y_v,SY_u)&= \frac{1}{4}\big ( g(\partial _r,SY_u)g(E_j,Y_v) - g(\partial _r,Y_v)g(E_j,SY_u)\\&\quad + g(\partial _r,JSY_u)g(E_j,JY_v) - g(\partial _r,JY_v)g(E_j,JSY_u)\\&\quad + 2g(\partial _r,J\!E_j)g(SY_u,JY_v)\big ) \\&= \frac{1}{4}(g(\partial _r,JSY_u)g(E_j,JY_v) - g(\partial _r,Y_v)g(E_j,JSY_u))\\&= \frac{1}{4}g(SY_u,J\!\partial _r) - \frac{1}{4}g(SY_u,J\!E_j). \end{aligned} \end{aligned}$$

Similarly, it holds that

$$\begin{aligned} \begin{aligned} R^0(\partial _r,Y_v,SY_u,E_j)&= \frac{1}{4}\big ( g(\partial _r,E_j)g(Y_v,SY_u) - g(\partial _r,SY_u)g(Y_v,E_j)\\&\quad + g(\partial _r,J\!E_j)g(Y_v,JSY_u) - g(\partial _r,JSY_u)g(Y_v,J\!E_j)\\&\quad + 2g(\partial _r,JY_v)g(E_j,JSY_u)\big ) \\&= -\frac{1}{4}g(\partial _r,JSY_u)g(Y_v,J\!E_j) + \frac{1}{2}g(\partial _r,JY_v)g(E_j,JSY_u)\\&= \frac{1}{4}g(SY_u,J\!\partial _r)g(Y_v,J\!E_j) +\frac{1}{2}g(SY_u,J\!E_j)g(Y_v,J\!E_j). \end{aligned} \end{aligned}$$

Since in addition $\partial _r\perp J\!\partial _r$, it follows that

$$\begin{aligned} \begin{aligned} R^0(\partial _r,\nabla _{Y_v}Y_u,\partial _r,E_j)&= \frac{1}{4}\big ( g(\partial _r,E_j)g(\nabla _{Y_v}Y_u,\partial _r) - g(\partial _r,\partial _r)g(\nabla _{Y_v}Y_u,E_j)\\&\quad + g(\partial _r,J\!E_j)g(\nabla _{Y_v}Y_u,J\!\partial _r) - g(\partial _r,J\!\partial _r)g(\nabla _{Y_v}Y_u,J\!E_j)\\&\quad + 2g(\partial _r,J\nabla _{Y_v}Y_u)g(E_j,J\!\partial _r)\big ) \\&= -\frac{1}{4}g(\nabla _{Y_v}Y_u,E_j). \end{aligned} \end{aligned}$$

Finally, since J is parallel and $g(\nabla _{Y_u}J\!E_j,\partial _r) = -g(SY_u,J\!E_j)$, it holds that

$$\begin{aligned} \begin{aligned} R^0(\partial _r,Y_v,\partial _r,\nabla _{Y_u}E_j)&= \frac{1}{4}\big ( g(\partial _r,\nabla _{Y_u}E_j)g(Y_v,\partial _r) - g(\partial _r,\partial _r)g(Y_v,\nabla _{Y_u}E_j)\\&\quad + g(\partial _r,J\nabla _{Y_u}E_j)g(Y_v,J\!\partial _r) - g(\partial _r,J\!\partial _r)g(Y_v,J\nabla _{Y_u}E_j)\\&\quad + 2g(\partial _r,JY_v)g(\nabla _{Y_u}E_j,J\!\partial _r)\big ) \\&= -\frac{1}{4} g(Y_v,\nabla _{Y_u}E_j) +\frac{1}{4}g(\partial _r,J\nabla _{Y_u}E_j)g(Y_v,J\!\partial _r)\\&\quad + \frac{1}{2}g(\partial _r,JY_v)g(\nabla _{Y_v}E_j,J\!\partial _r)\\&= -\frac{1}{4}g(Y_v,\nabla _{Y_u}E_j) + \frac{3}{4}g(\nabla _{Y_u}J\!E_j,\partial _r)g(Y_v,J\!\partial _r)\\&= -\frac{1}{4}g(Y_v,\nabla _{Y_u}E_j) - \frac{3}{4}g(SY_u,J\!E_j)g(Y_v,J\!\partial _r). \end{aligned} \end{aligned}$$

$\square$