1 Introduction
Let 
$(M,g)$ and 
$(N,h)$ be two Riemannian manifolds. The Dirichlet energy of a map 
$\unicode[STIX]{x1D711}:(M,g)\rightarrow (N,h)$ is defined as
$$\begin{eqnarray}E(\unicode[STIX]{x1D711})=\frac{1}{2}\int _{M}\Vert T\unicode[STIX]{x1D711}\Vert _{g,h}^{2}d\text{vol}_{g}.\end{eqnarray}$$ When 
$\dim M=2$, the energy is conformally invariant with respect to 
$g$. This is of considerable usefulness, for example, to construct conformal minimal immersions of Riemann surfaces [Reference MilnorMil79]. However, in higher dimension, the energy is no longer conformally invariant.
Critical points of a functional are solutions to a partial differential equation called the Euler–Lagrange equation of the functional; in other words, they form the kernel of a certain differential operator. In our case, the critical points of the Dirichlet energy are called harmonic maps, and harmonic functions 
$\unicode[STIX]{x1D711}:(M,g)\rightarrow (\mathbb{R},\text{eucl})$ coincide with the kernel of the Laplacian.
In a recent work, Bérard has shown the existence, given two Riemannian manifolds 
$(M,g)$ and 
$(N,h)$, with 
$M$ of even dimension 
$n$, of a functional 
${\mathcal{E}}_{g}^{n}$ on 
$C^{\infty }(M,N)$, conformally invariant with respect to 
$g$, and equal to the usual energy when 
$n=2$ [Reference BérardBér13]. This functional is called renormalized energy, and its critical points are called conformal-harmonic maps. Conformal-harmonic maps generalize harmonic maps; moreover, when 
$n=4$ and 
$N=\mathbb{R}$, the induced operator coincides with the Paneitz operator.
We develop here the notions of Cauchy–Riemann (CR)-harmonicity and renormalized energy in CR geometry. CR-harmonic maps also generalize CR-holomorphic maps, which are notoriously hard to come by. When 
$\dim M=3$ and 
$N=\mathbb{R}$, the induced operator coincides with the CR Paneitz operator. This generalizes the recent work of Marugame [Reference MarugameMar18]. Another extension of the CR Paneitz operator to maps has been proposed by Chong, Dong, Ren and Yang [Reference Chong, Dong, Ren and YangCDRY19]. The main result is the following, which summarizes Proposition 4.1 and Theorem 3.3.
Theorem 1.1. Let 
$(M^{2n+1},H,J,\unicode[STIX]{x1D703})$ be a compact strictly pseudoconvex pseudo-Hermitian manifold and 
$(N,h)$ be a Riemannian manifold. There exists a functional 
$F_{n}$ on 
$C^{\infty }(M,N)$ which is a CR invariant, that is, conformally invariant with respect to 
$\unicode[STIX]{x1D703}$. For 
$\unicode[STIX]{x1D711}\in C^{\infty }(M,N)$, it reads
$$\begin{eqnarray}\displaystyle F_{n}(\unicode[STIX]{x1D711}) & = & \displaystyle \frac{(-1)^{n+1}}{2n!^{2}}\int _{M}\left\langle (\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h})^{n-1}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}\right\rangle _{h}\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}\nonumber\\ \displaystyle & & \displaystyle +\,\text{lower order terms (in derivatives of }\unicode[STIX]{x1D711}\text{)},\nonumber\end{eqnarray}$$ where 
$\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}$ is the Webster divergence on 
$\unicode[STIX]{x1D6FA}^{1}(M)\otimes \unicode[STIX]{x1D711}^{\ast }TN$.
The Euler–Lagrange equation of 
$F_{n}$ is a partial differential equation of order 
$2n+2$, which is itself CR covariant. For 
$\unicode[STIX]{x1D711}\in C^{\infty }(M,N)$, it reads
$$\begin{eqnarray}0=\frac{(-1)^{n}}{n!}(\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h})^{n}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}+\text{lower order terms (in derivatives of }\unicode[STIX]{x1D711}\text{)}.\end{eqnarray}$$ Moreover, we provide explicit computations of 
$P_{1}$ and 
$F_{1}$ in Theorems 3.11 and 4.4, respectively.
The paper is organized as follows: in Section 2, we recall notions of asymptotically complex hyperbolic (ACH) geometry. In Section 3, we adapt the classical construction by Graham, Jenne, Mason and Sparling to obtain a CR Paneitz operator acting on maps, and we define CR-harmonicity [Reference Graham, Jenne, Mason and SparlingGJMS92]. We also provide an explicit computation of the operator in dimension 
$3$. In Section 4, we develop the corresponding notion of renormalized energy. Section 5 presents computations in higher dimension, which do not allow for an explicit expression of the operator. Finally, Section 6 gives a correspondence between CR-harmonic maps on a pseudo-Hermitian manifold and conformal-harmonic maps on its Fefferman bundle.
We adopt the following convention: small Greek letters will denote indices in 
$\{1,\ldots ,n\}$; capital Greek letters in 
$\{1,\ldots ,n,\overline{1},\ldots ,\overline{n}\}$; small Latin letters in 
$\{0,1,\ldots ,n\}$; capital Latin letters in 
$\{0,1,\ldots ,n,\overline{0},\overline{1},\ldots ,\overline{n}\}$. Moreover, we use the Einstein summation convention everywhere.
2 ACHE manifolds
Asymptotically hyperbolic (AH for short) manifolds are manifolds which admit a conformal infinity, that is to say, a boundary equipped with a conformal structure which is, roughly speaking, a generalization of the standard conformal sphere seen as the boundary of the Poincaré disk. Reciprocally, every compact conformal manifold can be filled with an AH manifold 
$X^{n+1}$ whose metric is Einstein, thus called AH-Einstein or AHE, when 
$n$ is odd. When 
$n$ is even, a conformally invariant obstruction to the existence of an AHE metric that is smooth up to the boundary appears [Reference Fefferman and GrahamFG85, Reference Graham, Hirachi and BiquardGH05]. Recently, Gursky and Székelyhidi have announced that an AHE metric exists locally for all 
$n\geqslant 3$ [Reference Gursky and SzékelyhidiGS17]. This approach provides a correspondence between a Riemannian structure on a manifold and a conformal structure on its boundary. Information on the conformal infinity can thus be read on the AHE metric.
The complex counterparts of AH manifolds, ACH manifolds, have been introduced by Epstein, Melrose and Mendoza [Reference Epstein, Melrose and MendozaEMM91]. They generalize the construction by Fefferman, Cheng and Yau, of asymptotically Bergman metrics, which are Kähler–Einstein metrics on bounded strictly pseudoconvex domains of 
$\mathbb{C}^{n+1}$, which are asymptotic to the CR structure of the boundary [Reference FeffermanFef76, Reference Cheng and YauCY80]. The regularity of these metrics near the boundary has been studied by Lee and Melrose [Reference Lee and MelroseLM82]. To an ACH manifold thus corresponds a CR infinity. For example, the CR infinity of the complex hyperbolic space 
$\mathbb{C}\mathbf{H}^{n+1}$ is 
$\mathbb{S}^{2n+1}$ endowed with its standard CR structure.
Because of the anisotropy of their structure, pseudo-Hermitian manifolds of odd dimension 
$N$ often behave, mutatis mutandis, like Riemannian manifolds of dimension 
$N+1$. They are sometimes said to have homogeneous dimension 
$N+1$ [Reference Jerison and LeeJL89]. In particular, ACH manifolds have been known to share similarities with the “
$n$ even” real case. The asymptotic development of ACH-Einstein (ACHE) and ACH-Kähler–Einstein metrics has been extensively studied by O. Biquard, M. Herzlich and Y. Matsumoto, and obstructions to smoothness have been identified [Reference BiquardBiq00, Reference Biquard and HerzlichBH05, Reference MatsumotoMat14].
Let us consider the sphere 
$\mathbb{S}^{2n+1}\subset \mathbb{C}^{n+1}$ endowed with its standard contact form
$$\begin{eqnarray}\unicode[STIX]{x1D703}_{0}=\frac{i}{4}\left(z_{j}d\overline{z}^{j}-\overline{z}_{j}dz^{j}\right)|_{\mathbb{ S}^{2n+1}}.\end{eqnarray}$$ Let 
$\unicode[STIX]{x1D6FE}_{0}=d\unicode[STIX]{x1D703}_{0}(\cdot ,i\cdot )$ be the induced metric on the contact distribution 
$\ker \unicode[STIX]{x1D703}_{0}$. The Bergman metric on the ball 
$\mathbb{B}^{2n+2}$ is given in polar coordinates by
$$\begin{eqnarray}g_{0}=dt^{2}+4\sinh ^{2}(t)\unicode[STIX]{x1D703}_{0}^{2}+4\sinh ^{2}\left(\frac{t}{2}\right)\unicode[STIX]{x1D6FE}_{0}.\end{eqnarray}$$ This metric is Kähler and has constant holomorphic sectional curvature 
$-1$. The space 
$(\mathbb{B}^{2n+2},g_{0})$ is known as the complex hyperbolic space and is denoted by 
$\mathbb{C}\mathbf{H}^{n+1}$.
More generally, let 
$(M,H,J)$ be a 
$(2n+1)$-dimensional orientable compact strictly pseudoconvex CR manifold. Namely, 
$H$ is an orientable hyperplane distribution in 
$TM$ and 
$J$ is a complex structure on 
$H$. Let 
$\unicode[STIX]{x1D703}$ be a compatible positive contact form and 
$\unicode[STIX]{x1D6FE}=d\unicode[STIX]{x1D703}(\cdot ,J\cdot )$ be the induced metric. Let 
$R$ be the Reeb field. Let 
$\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D703}}$ be the Tanaka–Webster connection of 
$(M,H,J,\unicode[STIX]{x1D703})$ and 
$\unicode[STIX]{x1D70F}$ be the pseudo-Hermitian torsion. Let 
$\overline{X}=[0,\unicode[STIX]{x1D700})\times M$, 
$\unicode[STIX]{x1D70B}:\overline{X}\rightarrow M$ be the natural projection and 
$r$ be the coordinate on 
$[0,\unicode[STIX]{x1D700})$. Let 
$X$ be the interior of 
$\overline{X}$. Let 
$g_{0}$ be the metric on 
$X$
$$\begin{eqnarray}g_{0}=\frac{dr^{2}}{r^{2}}+\frac{\unicode[STIX]{x1D703}^{2}}{r^{2}}+\frac{\unicode[STIX]{x1D6FE}}{r}.\end{eqnarray}$$ A function 
$s\in C^{\infty }(\overline{X},\mathbb{R}_{+})$ is called boundary defining if 
$s>0$ on 
$X$, 
$s=0$ and 
$ds\neq 0$ on 
$\{0\}\times M$. Equivalently, 
$s=e^{f}r$ for some 
$f$ in 
$C^{\infty }(\overline{X},\mathbb{R})$. A conformal change of the boundary defining function corresponds to a conformal change of the contact form. Indeed, let us consider 
$g_{0}$ as 
$g_{0}(r,\unicode[STIX]{x1D703})$, then, for 
$f$ in 
$C^{\infty }(\overline{X},\mathbb{R})$,
$$\begin{eqnarray}g_{0}(e^{f}r,\unicode[STIX]{x1D703})=g_{0}(r,e^{-f|_{M}}\unicode[STIX]{x1D703}).\end{eqnarray}$$ We define an order 
$O_{e}$ adapted to 
$g_{0}$. A normal basis with respect to 
$g_{0}$ is 
$e=(r\unicode[STIX]{x2202}_{r},rR,r^{1/2}T_{A})$, where 
$\left(T_{A}\right)$ is an orthonormal basis for 
$\unicode[STIX]{x1D6FE}$, considered as a Hermitian metric. Its dual basis is 
$e^{\ast }=(r^{-1}dr,r^{-1}\unicode[STIX]{x1D703},r^{-1/2}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FC}},r^{-1/2}\unicode[STIX]{x1D703}^{\overline{\unicode[STIX]{x1D6FC}}})$. The order 
$O_{e}$ takes 
$e$ and 
$e^{\ast }$ for reference. Thus, we have, for example,
$$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D703}^{\overline{\unicode[STIX]{x1D6FC}}}=r\big(r^{-1/2}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FC}}\big)\circ \big(r^{-1/2}\unicode[STIX]{x1D703}^{\overline{\unicode[STIX]{x1D6FC}}}\big)=O_{e}(r),\end{eqnarray}$$ where 
$\unicode[STIX]{x1D706}\circ \unicode[STIX]{x1D707}:=\unicode[STIX]{x1D706}\otimes \unicode[STIX]{x1D707}+\unicode[STIX]{x1D707}\otimes \unicode[STIX]{x1D706}$.
Definition 2.1. [Reference BiquardBiq00]
A metric 
$g$ on 
$X$ is said to be ACH if 
$g-g_{0}=o_{e}(1)$. The CR manifold 
$(M,H,J)$ is then called the CR infinity of 
$(X,g)$.
Example 2.2. For 
$\unicode[STIX]{x1D706}>0$,
$$\begin{eqnarray}g=\frac{dr^{2}}{r^{2}}+\frac{(1-\unicode[STIX]{x1D706}^{2}r^{2})^{2}}{r^{2}}\unicode[STIX]{x1D703}^{2}+\frac{(1-\unicode[STIX]{x1D706}r)^{2}}{r}\unicode[STIX]{x1D6FE}\end{eqnarray}$$ is an ACH metric on 
$X$. Moreover, if 
$(M,H,J,\unicode[STIX]{x1D703})$ is Einstein, that is, pseudo-Einstein with vanishing pseudo-Hermitian torsion, with 
$\text{Ric}_{W}(J,\unicode[STIX]{x1D703})=2(n+1)\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FE}$, then 
$g$ is an Einstein metric, satisfying
$$\begin{eqnarray}\text{Ric}(g)=-\frac{n+2}{2}g.\end{eqnarray}$$ Indeed, a complex structure 
$\tilde{J}$ compatible with 
$g$ on 
$X$ is given by 
$\tilde{J}|_{H\times \{r\}}=J$ and 
$\tilde{J}\unicode[STIX]{x2202}_{r}=-R/(1-\unicode[STIX]{x1D706}^{2}r^{2})$, that is, 
$dr\circ \tilde{J}=(1-\unicode[STIX]{x1D706}^{2}r^{2})\unicode[STIX]{x1D703}$. Let 
$\unicode[STIX]{x1D703}^{0}:=(1/\sqrt{2})(1/(1-\unicode[STIX]{x1D706}^{2}r^{2})dr-i\unicode[STIX]{x1D703})$ and let 
$\unicode[STIX]{x1D70E}:=\unicode[STIX]{x1D703}^{0}\wedge \unicode[STIX]{x1D703}^{1}\wedge \cdots \wedge \unicode[STIX]{x1D703}^{n}$ be a section of the canonical bundle.
Then
$$\begin{eqnarray}d\unicode[STIX]{x1D70E}=\frac{i}{\sqrt{2}}d\unicode[STIX]{x1D703}\wedge \unicode[STIX]{x1D703}^{1}\wedge \cdots \wedge \unicode[STIX]{x1D703}^{n}-\unicode[STIX]{x1D703}^{0}\wedge d\unicode[STIX]{x1D703}^{1}\wedge \cdots \wedge \unicode[STIX]{x1D703}^{n}+\cdots +(-1)^{n}\unicode[STIX]{x1D703}^{0}\wedge \unicode[STIX]{x1D703}^{1}\wedge \cdots \wedge d\unicode[STIX]{x1D703}^{n},\end{eqnarray}$$ where the first term vanishes, and since 
$\unicode[STIX]{x1D70F}=0$, 
$d\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FD}}\wedge \unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6FD}}^{\unicode[STIX]{x1D6FC}}$; hence,
$$\begin{eqnarray}d\unicode[STIX]{x1D70E}=-\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D6FC}}\wedge \unicode[STIX]{x1D70E}.\end{eqnarray}$$ The curvature form of 
$\unicode[STIX]{x1D70E}$, in the sense of [Reference Biquard and HerzlichBH05], is hence given by 
$-d\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D6FC}}=-{\mathcal{R}}^{\unicode[STIX]{x1D703}}\,_{\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D70C}}\,_{\unicode[STIX]{x1D70C}\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FC}}\wedge \unicode[STIX]{x1D703}^{\overline{\unicode[STIX]{x1D6FC}}}=2i(n+1)\unicode[STIX]{x1D706}d\unicode[STIX]{x1D703}$. Moreover,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}\wedge \overline{\unicode[STIX]{x1D70E}} & = & \displaystyle \frac{(-1)^{n+1}ir^{n+2}}{(1-\unicode[STIX]{x1D706}^{2}r^{2})^{2}(1-\unicode[STIX]{x1D706}r)^{2n}}(r^{-1}dr)\wedge ((1-\unicode[STIX]{x1D706}^{2}r^{2})r^{-1}\unicode[STIX]{x1D703})\wedge \left((1-\unicode[STIX]{x1D706}r)r^{-1/2}\unicode[STIX]{x1D703}^{1}\right)\nonumber\\ \displaystyle & & \displaystyle \quad \wedge \cdots \wedge \left((1-\unicode[STIX]{x1D706}r)r^{-1/2}\unicode[STIX]{x1D703}^{\overline{n}}\right).\nonumber\end{eqnarray}$$Consequently,
$$\begin{eqnarray}|\unicode[STIX]{x1D70E}|_{g}^{2}=\frac{r^{n+2}}{(1-\unicode[STIX]{x1D706}^{2}r^{2})^{2}(1-\unicode[STIX]{x1D706}r)^{2n}};\end{eqnarray}$$ hence, 
$\ln |\unicode[STIX]{x1D70E}|_{g}^{2}=(n+2)\ln r-2\ln (1+\unicode[STIX]{x1D706}r)-(2n+2)\ln (1-\unicode[STIX]{x1D706}r)$. We have
$$\begin{eqnarray}\unicode[STIX]{x2202}r={\textstyle \frac{1}{2}}(dr-i(1-\unicode[STIX]{x1D706}^{2}r^{2})\unicode[STIX]{x1D703});\end{eqnarray}$$hence,
$$\begin{eqnarray}i\overline{\unicode[STIX]{x2202}}\unicode[STIX]{x2202}r=-\unicode[STIX]{x1D706}^{2}rdr\wedge \unicode[STIX]{x1D703}+\frac{1-\unicode[STIX]{x1D706}^{2}r^{2}}{2}d\unicode[STIX]{x1D703}\qquad \text{and}\qquad i\overline{\unicode[STIX]{x2202}}r\wedge \unicode[STIX]{x2202}r=\frac{1-\unicode[STIX]{x1D706}^{2}r^{2}}{2}dr\wedge \unicode[STIX]{x1D703}.\end{eqnarray}$$ The Ricci form of 
$g$ is then given by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{g} & = & \displaystyle -i\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\ln |\unicode[STIX]{x1D70E}|_{g}^{2}+id\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6FC}}^{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & = & \displaystyle -i\unicode[STIX]{x2202}\overline{\unicode[STIX]{x2202}}\ln |\unicode[STIX]{x1D70E}|_{g}^{2}+2(n+1)\unicode[STIX]{x1D706}d\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & = & \displaystyle \frac{n+2}{2}\left(\frac{1-\unicode[STIX]{x1D706}^{2}r^{2}}{r^{2}}dr\wedge \unicode[STIX]{x1D703}-\frac{(1-\unicode[STIX]{x1D706}r)^{2}}{r}d\unicode[STIX]{x1D703}\right).\nonumber\end{eqnarray}$$ With this example in mind, one may ask if there is, in general, an ACHE metric on 
$X$. Contrarily to the theorem of Cheng–Yau for domains of 
$\mathbb{C}^{n+1}$, such a metric may not exist in general [Reference Cheng and YauCY80]. Nevertheless, there are formally determined almost ACHE metrics in the following sense.
Definition 2.3. In any asymptotic development 
$\sum _{k}a_{k}(p)r^{k}$, the term 
$a_{k}$, seen as a function on 
$M$, is called formally determined if it is a universal polynomial on a finite jet of the CR structure at 
$p\in M$ only.
Theorem 2.4. [Reference MatsumotoMat14]
There is an ACH metric 
$g_{E}$ on 
$X$, which is Einstein up to order 
$n+1$, that is,
$$\begin{eqnarray}\text{Ric}(g_{E})=-\frac{n+2}{2}g_{E}+O_{e}(r^{n+1}),\end{eqnarray}$$ where 
$O_{e}$ denotes the order with respect to any basis 
$e$ orthonormal for 
$g_{0}$. The metric 
$g_{E}$ is formally determined modulo 
$O_{e}(r^{n+1})$. Moreover, we have the asymptotic development
$$\begin{eqnarray}g_{E}=g_{0}+\unicode[STIX]{x1D6F7}+O_{e}(r^{3/2}),\end{eqnarray}$$where
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}=-2\text{Sch}_{W}(J,\unicode[STIX]{x1D703})+2\unicode[STIX]{x1D6FE}(J\unicode[STIX]{x1D70F}\cdot ,\cdot ),\end{eqnarray}$$where
$$\begin{eqnarray}\text{Sch}_{W}(J,\unicode[STIX]{x1D703})=\frac{1}{n+2}\left(\text{Ric}_{W}(J,\unicode[STIX]{x1D703})-\frac{\text{Scal}_{W}(J,\unicode[STIX]{x1D703})}{2(n+1)}\unicode[STIX]{x1D6FE}\right)\end{eqnarray}$$is the CR Schouten tensor.
Remark 2.5. Note that 
$\unicode[STIX]{x1D6F7}=O_{e}(r)$.
We thus have a formally determined almost ACHE metric on 
$X$. A more convenient metric for our study would be an almost ACH-Kähler–Einstein metric on 
$X$. We have at hand the following results.
Proposition 2.6. [Reference Biquard and HerzlichBH05]
One can construct on 
$X$ a formal complex structure 
$J_{X}$, entirely formally determined by the CR infinity, starting from the almost complex structure 
$\tilde{J}$, which is the extension of 
$J$ to 
$X$ with 
$\tilde{J}\unicode[STIX]{x2202}_{r}=R$. Moreover, an extension 
$\tilde{\unicode[STIX]{x1D6FB}}^{\unicode[STIX]{x1D703}}$ of 
$\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D703}}$ to 
$X$ is given by
$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D6FB}}^{\unicode[STIX]{x1D703}}r\unicode[STIX]{x2202}_{r}=\tilde{\unicode[STIX]{x1D6FB}}^{\unicode[STIX]{x1D703}}rR=\tilde{\unicode[STIX]{x1D6FB}}_{r\unicode[STIX]{x2202}_{r}}^{\unicode[STIX]{x1D703}}r^{1/2}T_{A}=0.\end{eqnarray}$$ Let 
$\tilde{T}^{\unicode[STIX]{x1D703}}$ be the torsion of 
$\tilde{\unicode[STIX]{x1D6FB}}^{\unicode[STIX]{x1D703}}$ and 
$\tilde{\unicode[STIX]{x1D70F}}:=\unicode[STIX]{x1D704}_{R}\tilde{T}^{\unicode[STIX]{x1D703}}$. An asymptotic development of 
$J_{X}$ is then given by
$$\begin{eqnarray}J_{X}=\tilde{J}-2r\tilde{\unicode[STIX]{x1D70F}}+O_{e}(r^{5/2}).\end{eqnarray}$$Theorem 2.7. [Reference FeffermanFef76, Reference Biquard and HerzlichBH05, Reference HerzlichHer07]
There is a formally determined ACH-Kähler metric 
$g_{KE}$ on 
$(X,J_{X})$, which is Einstein up to order 
$n+\frac{3}{2}$, that is,
$$\begin{eqnarray}\text{Ric}(g_{KE})=-\frac{n+2}{2}g_{KE}+O_{e}(r^{n+\frac{3}{2}}).\end{eqnarray}$$ Moreover, 
$g_{E}$ and 
$g_{KE}$ coincide up to order 
$n+\frac{1}{2}$.
In dimension 
$2n+1=3$, the asymptotic development of 
$g_{KE}$, and therefore of 
$g_{E}$, is known at order 
$\frac{3}{2}$, which will be essential in Sections 3.4 and 4.2.
Theorem 2.8. [Reference Biquard and HerzlichBH05, Reference HerzlichHer07]
When 
$n=1$, we have the asymptotic development
$$\begin{eqnarray}g_{KE}=g_{0}+\unicode[STIX]{x1D6F7}_{AB}\unicode[STIX]{x1D703}^{A}\circ \unicode[STIX]{x1D703}^{B}+\unicode[STIX]{x1D6F9}_{0\overline{1}}\unicode[STIX]{x1D703}^{0}\circ \unicode[STIX]{x1D703}^{\overline{1}}+\unicode[STIX]{x1D6F9}_{\overline{0}1}\unicode[STIX]{x1D703}^{\overline{0}}\circ \unicode[STIX]{x1D703}^{1}+O_{e}(r^{2}),\end{eqnarray}$$where
$$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{0\overline{1}}=-\sqrt{2}\left(\frac{1}{6}{\text{Scal}_{W}}_{,\overline{1}}-\frac{2i}{3}\unicode[STIX]{x1D70F}_{\overline{1},1}^{1}\right),\end{eqnarray}$$ and 
$\unicode[STIX]{x1D6F7}$ is given by Theorem 2.4:
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{1\overline{1}}=-\frac{\text{Scal}_{W}}{4}\qquad \text{and}\qquad \unicode[STIX]{x1D6F7}_{11}=-i\unicode[STIX]{x1D70F}_{1}^{\overline{1}}.\end{eqnarray}$$3 CR-harmonic maps
3.1 Definitions
Let 
$(M,H,J)$ be a 
$(2n+1)$-dimensional orientable, compact, strictly pseudoconvex CR manifold and 
$(X,g)$ be an ACH manifold with CR infinity 
$(M,H,J)$, where 
$g$ is the approximately ACH-Kähler–Einstein metric given by Theorem 2.7. Let 
$\unicode[STIX]{x1D70B}:X\rightarrow M$ be the standard projection. Let 
$(N,h)$ be a Riemannian manifold. Let 
$\unicode[STIX]{x1D711}\in C^{\infty }(M,N)$, and let 
$\tilde{\unicode[STIX]{x1D711}}\in C^{\infty }(\overline{X},N)$ be any extension of 
$\unicode[STIX]{x1D711}$, that is, 
$\tilde{\unicode[STIX]{x1D711}}|_{M}=\unicode[STIX]{x1D711}$.
Let 
$T\tilde{\unicode[STIX]{x1D711}}$ be the tangent map of 
$\tilde{\unicode[STIX]{x1D711}}$. It is a section of the bundle 
$\unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$, and its norm is defined by
$$\begin{eqnarray}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}:=\text{tr}_{g}(\tilde{\unicode[STIX]{x1D711}}^{\ast }h).\end{eqnarray}$$ The bundle 
$\unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$ is canonically equipped with the connection
$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{g,h}:=\unicode[STIX]{x1D6FB}^{g}\otimes 1_{\tilde{\unicode[STIX]{x1D711}}^{\ast }TN}+1_{\unicode[STIX]{x1D6FA}^{1}(\overline{X})}\otimes \unicode[STIX]{x1D6FB}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h},\end{eqnarray}$$ where 
$\unicode[STIX]{x1D6FB}^{g}$ and 
$\unicode[STIX]{x1D6FB}^{h}$ are the respective Levi-Civita connections of 
$g$ and 
$h$, and 
$\unicode[STIX]{x1D6FB}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}:=\tilde{\unicode[STIX]{x1D711}}^{\ast }\unicode[STIX]{x1D6FB}^{h}$.
The divergence 
$\unicode[STIX]{x1D6FF}^{g,h}$ is then defined for 
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$ by
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}:=-(\unicode[STIX]{x1D6FB}_{e_{I}}^{g,h}\unicode[STIX]{x1D714})(e_{\overline{I}}),\end{eqnarray}$$ where 
$(e_{i})$ is an orthonormal basis of 
$T^{1,0}\overline{X}$ for 
$g$, considered as a Hermitian metric. We thus have
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}=-\unicode[STIX]{x1D6FB}_{e_{I}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}(\unicode[STIX]{x1D714}(e_{\overline{I}}))+\unicode[STIX]{x1D714}(\unicode[STIX]{x1D6FB}_{e_{I}}^{g}e_{\overline{I}}).\end{eqnarray}$$ For 
$\unicode[STIX]{x1D70C}\in (0,\unicode[STIX]{x1D700})$, the energy of 
$\tilde{\unicode[STIX]{x1D711}}$ in 
$(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D700})\times M$ is the functional
$$\begin{eqnarray}E(\tilde{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D70C})=\frac{1}{2}\int _{(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D700})\times M}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}d\text{vol}_{g}.\end{eqnarray}$$ An extension 
$\tilde{\unicode[STIX]{x1D711}}$ is said to be harmonic if it is a critical point of the energy for all 
$\unicode[STIX]{x1D70C}$. Equivalently, 
$\tilde{\unicode[STIX]{x1D711}}$ is harmonic if and only if 
$\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}=0$.
Following the ideas of Graham, Jenne, Mason and Sparling, we want to find the obstructions to the existence of a smooth harmonic extension [Reference Graham, Jenne, Mason and SparlingGJMS92]. More precisely, assuming that 
$\tilde{\unicode[STIX]{x1D711}}$ is smooth, we want to know if the first terms of the asymptotic development of 
$\tilde{\unicode[STIX]{x1D711}}$ are determined by the data at infinity. By similarity with the real case and based on the known asymptotic developments of the approximately ACHE metrics, we expect to find an obstruction at order 
$n+1$, taking the form of a CR covariant differential operator of order 
$2n+2$.
Here, the asymptotic development of 
$\tilde{\unicode[STIX]{x1D711}}$ will denote, by identification, the asymptotic development in 
$r$ of 
$U:=\exp _{\unicode[STIX]{x1D711}}^{-1}\circ \tilde{\unicode[STIX]{x1D711}}\in C^{\infty }(\overline{X},(\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D70B})^{\ast }TN)$, that is,
$$\begin{eqnarray}\forall p\in M,~\forall r\in (0,\unicode[STIX]{x1D700}),\qquad \tilde{\unicode[STIX]{x1D711}}(p,r):=\exp _{\unicode[STIX]{x1D711}(p)}\left(U(p,r)\right),\end{eqnarray}$$ where, for 
$p\in M$, the exponential map 
$\exp _{\unicode[STIX]{x1D711}(p)}$ is a diffeomorphism between a small ball 
$B(0,\unicode[STIX]{x1D700})\subset T_{\unicode[STIX]{x1D711}(p)}N$ and its image, which is a neighborhood in 
$N$ of 
$\unicode[STIX]{x1D711}(p)$. Note that 
$U(\cdot ,0)=0$. We denote 
$v\tilde{\unicode[STIX]{x1D711}}:=T\tilde{\unicode[STIX]{x1D711}}(v)$ for 
$v\in TX$, and similarly for 
$\unicode[STIX]{x1D711}$ on 
$TM$, and
$$\begin{eqnarray}\forall k\geqslant 1,\quad \unicode[STIX]{x1D711}_{k}:=(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h})^{k-1}\unicode[STIX]{x2202}_{r}\tilde{\unicode[STIX]{x1D711}}|_{r=0}.\end{eqnarray}$$ Note that 
$\unicode[STIX]{x1D711}_{k}$ is a section of 
$\unicode[STIX]{x1D711}^{\ast }TN$; hence, 
$\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D711}_{k}$ is a section of 
$\unicode[STIX]{x1D6FA}^{1}(M)\otimes \unicode[STIX]{x1D711}^{\ast }TN$.
3.2 Computation of the divergence
We use the notations of Section 2. Let 
$(T_{\unicode[STIX]{x1D6FC}})$ be a local basis of 
$T^{1,0}M$ and 
$T_{\overline{\unicode[STIX]{x1D6FC}}}:=\overline{T_{\unicode[STIX]{x1D6FC}}}$ such that 
$(T_{A})$ is orthonormal for 
$\unicode[STIX]{x1D6FE}$, considered as a Hermitian metric. Let 
$(\unicode[STIX]{x1D703}^{A})$ be the basis dual to 
$(T_{A})$. Let 
$T_{0}:=(\unicode[STIX]{x2202}_{r}-iR)/\sqrt{2}$ and 
$\unicode[STIX]{x1D703}^{0}:=(dr+i\unicode[STIX]{x1D703})/\sqrt{2}$ be its dual.
Lemma 3.1. For 
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}^{g_{0},h}\unicode[STIX]{x1D714} & = & \displaystyle nr\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-r^{2}\left(\unicode[STIX]{x1D6FB}_{T_{0}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{0}})+\unicode[STIX]{x1D6FB}_{T_{\overline{0}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{0})\right)-r\unicode[STIX]{x1D6FB}_{T_{A}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{A}})\nonumber\\ \displaystyle & = & \displaystyle nr\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-r^{2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R)-r\unicode[STIX]{x1D6FB}_{T_{A}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{A}}).\nonumber\end{eqnarray}$$Proof. We have
$$\begin{eqnarray}g_{0}=r^{-2}\unicode[STIX]{x1D703}^{0}\circ \unicode[STIX]{x1D703}^{\overline{0}}+r^{-1}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FC}}\circ \unicode[STIX]{x1D703}^{\overline{\unicode[STIX]{x1D6FC}}}.\end{eqnarray}$$ An orthonormal basis of 
$T^{1,0}\overline{X}$ with respect to 
$g_{0}$ is hence given by
$$\begin{eqnarray}\displaystyle (e_{0}^{(0)},e_{\unicode[STIX]{x1D6FC}}^{(0)}):=(rT_{0},r^{1/2}T_{\unicode[STIX]{x1D6FC}}). & & \displaystyle \nonumber\end{eqnarray}$$ The trace of the Levi-Civita connection of 
$g_{0}$ is given in this basis by the Koszul formula:
$$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{e_{I}^{(0)}}^{g_{0}}e_{\overline{I}}^{(0)}=g_{0}([e_{\overline{J}}^{(0)},e_{I}^{(0)}],e_{\overline{I}}^{(0)})e_{J}^{(0)}.\end{eqnarray}$$ Let 
$\tilde{\unicode[STIX]{x1D6FB}}^{\unicode[STIX]{x1D703}}$ be the extension of 
$\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D703}}$ given by Proposition 2.6. We have
$$\begin{eqnarray}\displaystyle \,[e_{0}^{(0)},e_{\overline{0}}^{(0)}] & = & \displaystyle \frac{1}{\sqrt{2}}(e_{\overline{0}}^{(0)}-e_{0}^{(0)}),\nonumber\\ \displaystyle \,[e_{0}^{(0)},e_{A}^{(0)}] & = & \displaystyle \frac{1}{\sqrt{2}}\left(\frac{1}{2}e_{A}^{(0)}-i\left(\tilde{\unicode[STIX]{x1D6FB}}_{{e_{0}}^{(0)}}^{\unicode[STIX]{x1D703}}e_{A}^{(0)}-\unicode[STIX]{x1D70F}(e_{A}^{(0)})\right)\right),\nonumber\\ \displaystyle \,[e_{A}^{(0)},e_{B}^{(0)}] & = & \displaystyle rd\unicode[STIX]{x1D703}(T_{A},T_{B})R.\nonumber\end{eqnarray}$$ Then, since 
$\text{tr}(\unicode[STIX]{x1D70F})=0$,
$$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{e_{I}^{(0)}}^{g_{0}}e_{\overline{I}}^{(0)}=(n+1)r\unicode[STIX]{x2202}_{r},\end{eqnarray}$$and also,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{e_{0}^{(0)}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{0}}^{(0)})+\unicode[STIX]{x1D6FB}_{e_{\overline{0}}^{(0)}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{0}^{(0)}) & = & \displaystyle r\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+r^{2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FB}_{e_{\unicode[STIX]{x1D6FC}}^{(0)}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{\unicode[STIX]{x1D6FC}}}^{(0)}) & = & \displaystyle r\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}}).\nonumber\end{eqnarray}$$ Hence, we have the announced expression for 
$\unicode[STIX]{x1D6FF}^{g_{0},h}\unicode[STIX]{x1D714}$.◻
Let us denote by 
$(\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714})^{(1)}$ the remainder of 
$\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}$, that is,
$$\begin{eqnarray}(\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714})^{(1)}:=\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FF}^{g_{0},h}\unicode[STIX]{x1D714}.\end{eqnarray}$$We prove the following technical lemma, which is crucial for the proof of Theorem 3.3.
Lemma 3.2. For 
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$, denoting by 
$O_{T}$ the order with respect to the basis 
$(\tilde{\unicode[STIX]{x1D711}}_{\ast }T_{I})$ in powers of 
$r$, we have
$$\begin{eqnarray}(\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714})^{(1)}=O_{T}(r^{2}),\end{eqnarray}$$ and there is no term of order 
$2$ in the remainder of the form 
$r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})$.
Proof. By Theorem 2.4, we have
$$\begin{eqnarray}g-g_{0}=\unicode[STIX]{x1D6F7}+O_{e}(r^{3/2})=\unicode[STIX]{x1D6F7}_{AB}\unicode[STIX]{x1D703}^{A}\circ \unicode[STIX]{x1D703}^{B}+O_{e}(r^{3/2}),\end{eqnarray}$$ where we recall that 
$\unicode[STIX]{x1D6F7}=-2\text{Sch}_{W}(J,\unicode[STIX]{x1D703})+2\unicode[STIX]{x1D6FE}(J\unicode[STIX]{x1D70F}\cdot ,\cdot )$ and that 
$O_{e}$ denotes the order with respect to 
$(e_{I}^{(0)})$. Note that 
$\unicode[STIX]{x1D6F7}_{AB}=\unicode[STIX]{x1D6F7}_{BA}$. Since 
$\unicode[STIX]{x1D6F7}$ is real, we have also 
$\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FD}}=\overline{\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}}$ and 
$\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}}}=\overline{\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}}$.
An orthonormal basis of 
$T^{1,0}\overline{X}$ with respect to 
$g$ induced from 
$e^{(0)}$ is formally given by
$$\begin{eqnarray}\displaystyle (e_{0},e_{\unicode[STIX]{x1D6FC}}) & :=\left(e_{0}^{(0)}+e_{0}^{(1)},e_{\unicode[STIX]{x1D6FC}}^{(0)}+e_{\unicode[STIX]{x1D6FC}}^{(1)}\right), & \displaystyle \nonumber\end{eqnarray}$$ where, by the Gram–Schmidt process, and since 
$\unicode[STIX]{x1D6F7}$ is horizontal,
$$\begin{eqnarray}e_{0}^{(1)}=O_{e}(r^{3/2})\qquad \text{and}\qquad e_{\unicode[STIX]{x1D6FC}}^{(1)}=O_{e}(r).\end{eqnarray}$$This leads to
$$\begin{eqnarray}\displaystyle \big(\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}\big)^{(1)} & = & \displaystyle -\unicode[STIX]{x1D6FB}_{e_{I}^{(0)}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}\big(e_{\overline{I}}^{(1)}\big)-\unicode[STIX]{x1D6FB}_{e_{I}^{(1)}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}\big(e_{\overline{I}}^{(0)}\big)-\unicode[STIX]{x1D6FB}_{e_{I}^{(1)}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}\big(e_{\overline{I}}^{(1)}\big)\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D714}\Big(\unicode[STIX]{x1D6FB}_{e_{I}^{(0)}}^{g}e_{\overline{I}}^{(0)}-\unicode[STIX]{x1D6FB}_{e_{I}^{(0)}}^{g_{0}}e_{\overline{I}}^{(0)}\Big)+\unicode[STIX]{x1D714}\Big(\unicode[STIX]{x1D6FB}_{e_{I}^{(0)}}^{g}e_{\overline{I}}^{(1)}\Big)+\unicode[STIX]{x1D714}\Big(\unicode[STIX]{x1D6FB}_{e_{I}^{(1)}}^{g}e_{\overline{I}}^{(0)}\Big)+\unicode[STIX]{x1D714}\Big(\unicode[STIX]{x1D6FB}_{e_{I}^{(1)}}^{g}e_{\overline{I}}^{(1)}\Big)\!,\nonumber\end{eqnarray}$$ all terms of which are in 
$O_{T}(r^{2})$ and are not of the form 
$r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})$.◻
3.3 An obstruction to regularity
Theorem 3.3. Let 
$(M,H,J)$ be a 
$(2n+1)$-dimensional orientable, compact, strictly pseudoconvex CR manifold and 
$(X,g)$ be an ACH manifold with CR infinity 
$(M,H,J)$, where 
$g$ is the approximately ACH-Kähler–Einstein metric given by Theorem 2.7. Let 
$\unicode[STIX]{x1D70B}:\overline{X}\rightarrow M$ be the standard projection. Let 
$(N,h)$ be a Riemannian manifold, and let 
$\unicode[STIX]{x1D711}\in C^{\infty }(M,N)$.
There exists a section 
$U$ of 
$(\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D70B})^{\ast }TN$, unique modulo 
$O_{T}(r^{n+1})$, such that 
$\tilde{\unicode[STIX]{x1D711}}=\exp _{\unicode[STIX]{x1D711}}\circ U$ satisfies
$$\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}rcl@{}}\tilde{\unicode[STIX]{x1D711}}|_{M} & = & \unicode[STIX]{x1D711},\\ \unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}} & = & O_{T}(r^{n+2}).\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$ The asymptotic development in 
$r$ of 
$U$ is
$$\begin{eqnarray}U=U_{1}r+\cdots +U_{n}\frac{r^{n}}{n!}+P_{n}(\unicode[STIX]{x1D711})\frac{r^{n+1}}{(n+1)!}\log r+O_{T}(r^{n+1}),\end{eqnarray}$$ where 
$U_{1},\ldots ,U_{n},P_{n}$ are formally determined by 
$\unicode[STIX]{x1D711}$, 
$g$ and 
$h$.
$P_{n}(\unicode[STIX]{x1D711})$ is an obstruction to the regularity of 
$U$ and is given by
$$\begin{eqnarray}\displaystyle P_{n}(\unicode[STIX]{x1D711}) & = & \displaystyle \left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}-n\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n-1}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}R\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{n+1}\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n+1}\left(\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}\right)^{(1)}\right|_{r=0}\nonumber\\ \displaystyle & = & \displaystyle \frac{(-1)^{n}}{n!}(\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h})^{n}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}+\text{lower order terms (in derivatives of }\unicode[STIX]{x1D711}\text{)}.\nonumber\end{eqnarray}$$Proof. For 
$m\in \mathbb{N}$, we have
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}=O_{T}(r^{m+1})\quad \Longleftrightarrow \quad \forall k\leqslant m,\quad \left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k}\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}=0.\end{eqnarray}$$We recall the notation
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{k}:=\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k-1}\unicode[STIX]{x2202}_{r}\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}.\end{eqnarray}$$ Now, by Lemma 3.1, we have, for 
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$,
$$\begin{eqnarray}\left.\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}\right|_{r=0}=n\left.\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})\right|_{r=0}+\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}(\unicode[STIX]{x1D714}|_{r=0}),\end{eqnarray}$$ and, for all 
$2\leqslant k\leqslant n$,
$$\begin{eqnarray}\displaystyle \frac{1}{k}\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k}\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}\right|_{r=0} & = & \displaystyle (n-k+1)\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k-1}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})\right|_{r=0}+\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k-1}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D714}\right|_{r=0}\nonumber\\ \displaystyle & & \displaystyle -\,(k-1)\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k-2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R)\right|_{r=0}+\frac{1}{k}\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k}\left(\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}\right)^{(1)}\right|_{r=0},\nonumber\end{eqnarray}$$where
$$\begin{eqnarray}\forall \unicode[STIX]{x1D714}_{0}\in \unicode[STIX]{x1D6FA}^{1}(M)\otimes \unicode[STIX]{x1D711}^{\ast }TN,\qquad \unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D714}_{0}:=-\unicode[STIX]{x1D6FB}_{T_{A}}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D714}_{0}(T_{\overline{A}})\end{eqnarray}$$and
$$\begin{eqnarray}\forall \unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN,\qquad \tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D714}:=-\unicode[STIX]{x1D6FB}_{T_{A}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{A}}).\end{eqnarray}$$ Consequently, 
$\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}=O_{T}(r^{n+1})$ is equivalent to
$$\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}rcl@{}}n\unicode[STIX]{x1D711}_{1} & = & -\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711},\\ (n-k+1)\unicode[STIX]{x1D711}_{k} & = & -\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k-1}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}-D_{k-1}(\unicode[STIX]{x1D711})\quad \forall 2\leqslant k\leqslant n,\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$where
$$\begin{eqnarray}D_{k-1}(\unicode[STIX]{x1D711}):=-(k-1)\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k-2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}R\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}+\frac{1}{k}\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{k}\left(\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}\right)^{(1)}\right|_{r=0}.\end{eqnarray}$$ By Lemma 3.2, 
$D_{k-1}(\unicode[STIX]{x1D711})$ only depends on 
$\unicode[STIX]{x1D711},\unicode[STIX]{x1D711}_{1},\ldots ,\unicode[STIX]{x1D711}_{k-1}$. This observation comes from the fact that, although
$$\begin{eqnarray}\displaystyle & \displaystyle \left.\big(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big)^{k}\left(r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x2202}_{r}\tilde{\unicode[STIX]{x1D711}}\right)\right|_{r=0}=2\unicode[STIX]{x1D711}_{k}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \forall X,Y\in \{\unicode[STIX]{x2202}_{r},R,T_{A}\},\quad \!\!(X,Y)\neq \!(\unicode[STIX]{x2202}_{r},\unicode[STIX]{x2202}_{r}),\quad \left.\!\!\big(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big)^{k}\!\!\big(r^{2}\unicode[STIX]{x1D6FB}_{X}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}Y\tilde{\unicode[STIX]{x1D711}}\big)\!\right|_{r=0}\!\text{does not depend on }\unicode[STIX]{x1D711}_{k}. & \displaystyle \nonumber\end{eqnarray}$$ By induction, 
$D_{k-1}(\unicode[STIX]{x1D711})$ is thus well defined.
In conclusion, requiring 
$\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}=O_{T}(r^{n+1})$ gives an asymptotic development for 
$\tilde{\unicode[STIX]{x1D711}}$ in powers of 
$r$, and this development is unique up to order 
$n$ with respect to 
$T$.
Assume now that 
$\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}=O_{T}(r^{n+1})$ and that 
$\tilde{\unicode[STIX]{x1D711}}$ admits a Taylor development up to order 
$n+1$. Then
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}=O_{T}(r^{n+2})\Longleftrightarrow \left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}+D_{n}(\unicode[STIX]{x1D711})=0.\end{eqnarray}$$ This equality cannot be true in general. Consequently, we introduce a term in 
$r^{n+1}\log r$ in the development of 
$\tilde{\unicode[STIX]{x1D711}}$:
$$\begin{eqnarray}U=U_{1}r+\cdots +U_{n}\frac{r^{n}}{n!}+P_{n}(\unicode[STIX]{x1D711})\frac{r^{n+1}}{(n+1)!}\log r+O_{T}(r^{n+1}).\end{eqnarray}$$ The coefficient 
$P_{n}(\unicode[STIX]{x1D711})$ verifies
$$\begin{eqnarray}\frac{1}{n+1}\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n+1}\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}=-P_{n}(\unicode[STIX]{x1D711})+\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}+D_{n}(\unicode[STIX]{x1D711});\end{eqnarray}$$hence,
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}=O_{T}(r^{n+2})\Longleftrightarrow P_{n}(\unicode[STIX]{x1D711})=\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}+D_{n}(\unicode[STIX]{x1D711}).\end{eqnarray}$$ This yields the announced obstruction, which only depends on 
$\unicode[STIX]{x1D711}$. Since
$$\begin{eqnarray}\unicode[STIX]{x1D711}_{k}=-\frac{1}{n-k+1}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D711}_{k-1}+\text{lower order terms (in derivatives of }\unicode[STIX]{x1D711}\text{)},\end{eqnarray}$$we have the announced leading term. ◻
Proposition 3.4. 
$P_{n}$ does not depend on whether we take 
$g=g_{E}$ or 
$g_{KE}$ on 
$X$.
Proof. To compute 
$P_{n}$, it is sufficient to be able to compute
$$\begin{eqnarray}\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{n+1}\left(\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}\right)^{(1)}\right|_{r=0};\end{eqnarray}$$ that is, by the proof of Lemma 3.2, to know the 
$e_{I}^{(1)}$ at order 
$n+1/2$ with respect to 
$e^{(0)}$. By the Gram–Schmidt process, it is thus sufficient to know 
$g$ at order 
$n+1/2$ with respect to 
$e^{(0)}$. Hence, by Theorems 2.4 and 2.7, we can equivalently consider 
$g_{E}$ or 
$g_{KE}$.◻
Proposition 3.5. Let 
$f\in C^{\infty }(\overline{X},\mathbb{R})$ and 
$f_{0}:=f|_{M}$, and let 
$\hat{r}=e^{f}r$ be a conformal change of boundary defining function. Then
$$\begin{eqnarray}\hat{P}_{n}(\unicode[STIX]{x1D711})=e^{-(n+1)f_{0}}P_{n}(\unicode[STIX]{x1D711}).\end{eqnarray}$$ The obstruction 
$P_{n}(\unicode[STIX]{x1D711})$ to the regularity of 
$\tilde{\unicode[STIX]{x1D711}}$ is therefore CR covariant.
Proof. We have
$$\begin{eqnarray}U=U_{1}r+\cdots +U_{n}\frac{r^{n}}{n!}+P_{n}(\unicode[STIX]{x1D711})\frac{r^{n+1}}{(n+1)!}\log r+O_{T}(r^{n+1}).\end{eqnarray}$$ Now, since 
$\exp _{\unicode[STIX]{x1D711}}:(\unicode[STIX]{x1D711}\circ \unicode[STIX]{x1D70B})^{\ast }TN\rightarrow N$ does not depend on 
$r$, neither does 
$U$. Moreover, since 
$M$ is compact, 
$\forall k,~O_{T}(\hat{r}^{k})=O_{T}(r^{k})$. We thus have
$$\begin{eqnarray}\displaystyle U & = & \displaystyle \hat{U} _{1}\hat{r}+\cdots +\hat{U} _{n}\frac{\hat{r}^{n}}{n!}+\hat{P}_{n}(\unicode[STIX]{x1D711})\frac{\hat{r}^{n+1}}{(n+1)!}\log \hat{r}+O_{T}(\hat{r}^{n+1})\nonumber\\ \displaystyle & = & \displaystyle \hat{U} _{1}e^{f}r+\cdots +\hat{U} _{n}e^{nf}\frac{r^{n}}{n!}+\hat{P}_{n}(\unicode[STIX]{x1D711})e^{(n+1)f}\frac{r^{n+1}}{(n+1)!}\log r+O_{T}(r^{n+1}).\nonumber\end{eqnarray}$$ Since the function 
$f$ itself has a Taylor expansion in 
$r$, all polynomial terms are mixed. However, there is only one term with order 
$r^{n+1}\log r$. By identification, this yields the result.◻
We then introduce CR-harmonic maps as maps for which the obstruction vanishes.
Definition 3.6. If 
$P_{n}(\unicode[STIX]{x1D711})=0$, 
$\unicode[STIX]{x1D711}$ is said to be CR-harmonic.
Example 3.7. Let us assume that 
$(M,H,J,\unicode[STIX]{x1D703})$ is Einstein with 
$\text{Ric}_{W}=2\unicode[STIX]{x1D706}(n+1)\unicode[STIX]{x1D6FE}$. We know from Example 2.2 that
$$\begin{eqnarray}g=\frac{dr^{2}}{r^{2}}+\frac{(1-\unicode[STIX]{x1D706}^{2}r^{2})^{2}}{r^{2}}\unicode[STIX]{x1D703}^{2}+\frac{(1-\unicode[STIX]{x1D706}r)^{2}}{r}\unicode[STIX]{x1D6FE}\end{eqnarray}$$ satisfies 
$\text{Ric}(g)=-((n+2)/2)g$. In this case, we can explicitly compute the divergence 
$\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}$, for 
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{1}(\overline{X})\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$.
Indeed, an orthonormal basis of 
$T^{1,0}\overline{X}$ with respect to 
$g$ induced from 
$e^{(0)}$ is given by
$$\begin{eqnarray}\displaystyle (e_{0},e_{\unicode[STIX]{x1D6FC}}):=\left(\frac{1}{\sqrt{2}}\left(r\unicode[STIX]{x2202}_{r}-i\frac{r}{1-\unicode[STIX]{x1D706}^{2}r^{2}}R\right),\frac{r^{1/2}}{1-\unicode[STIX]{x1D706}r}T_{\unicode[STIX]{x1D6FC}}\right); & & \displaystyle \nonumber\end{eqnarray}$$hence,
$$\begin{eqnarray}\displaystyle \,[e_{0},e_{\overline{0}}] & = & \displaystyle \frac{1}{\sqrt{2}}\frac{1+\unicode[STIX]{x1D706}^{2}r^{2}}{1-\unicode[STIX]{x1D706}^{2}r^{2}}\left(e_{\overline{0}}-e_{0}\right),\nonumber\\ \displaystyle \,[e_{0},e_{A}] & = & \displaystyle \frac{1}{2\sqrt{2}}\frac{1+\unicode[STIX]{x1D706}r}{1-\unicode[STIX]{x1D706}r}e_{A},\nonumber\\ \displaystyle \,[e_{A},e_{B}] & = & \displaystyle \frac{r}{(1-\unicode[STIX]{x1D706}r)^{2}}d\unicode[STIX]{x1D703}(T_{A},T_{B})R.\nonumber\end{eqnarray}$$Then
$$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{e_{I}}^{g}e_{\overline{I}}=\left(n\frac{1+\unicode[STIX]{x1D706}^{2}r^{2}}{1-\unicode[STIX]{x1D706}^{2}r^{2}}+\frac{1+\unicode[STIX]{x1D706}r}{1-\unicode[STIX]{x1D706}r}\right)r\unicode[STIX]{x2202}_{r},\end{eqnarray}$$and also,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{e_{0}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{0}})+\unicode[STIX]{x1D6FB}_{e_{\overline{0}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{0}) & = & \displaystyle r\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+\frac{r^{2}}{(1-\unicode[STIX]{x1D706}^{2}r^{2})^{2}}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FB}_{e_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{\unicode[STIX]{x1D6FC}}}) & = & \displaystyle \frac{r}{(1-\unicode[STIX]{x1D706}r)^{2}}\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}}).\nonumber\end{eqnarray}$$The divergence is hence given by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714} & = & \displaystyle \left(n\frac{1+\unicode[STIX]{x1D706}^{2}r^{2}}{1-\unicode[STIX]{x1D706}^{2}r^{2}}+\frac{1+\unicode[STIX]{x1D706}r}{1-\unicode[STIX]{x1D706}r}-1\right)r\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})\nonumber\\ \displaystyle & & \displaystyle -\,\frac{r^{2}}{(1-\unicode[STIX]{x1D706}^{2}r^{2})^{2}}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R)+\frac{r}{(1-\unicode[STIX]{x1D706}r)^{2}}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D714}.\nonumber\end{eqnarray}$$From Example 3.7, we get the following results.
Corollary 3.8. If 
$(M,H,J,\unicode[STIX]{x1D703})$ is Einstein, then subharmonic maps which verify 
$\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}=0$ are CR-harmonic.
Proof. Indeed, let 
$\unicode[STIX]{x1D711}$ be subharmonic, that is, 
$\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}=0$, and such that 
$\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}=0$. Let 
$\tilde{\unicode[STIX]{x1D711}}$ be the extension of 
$\unicode[STIX]{x1D711}$ given by Theorem 3.3. We thus have 
$\unicode[STIX]{x1D711}_{1}=0$. Moreover, by Example 3.7, we have
$$\begin{eqnarray}(\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}})^{(1)}=\unicode[STIX]{x1D6FC}(r)\unicode[STIX]{x2202}_{r}\tilde{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D6FD}(r)\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}R\tilde{\unicode[STIX]{x1D711}}+\unicode[STIX]{x1D6FE}(r)\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}},\end{eqnarray}$$ where 
$\unicode[STIX]{x1D6FC}(r)=O(r^{2})$, 
$\unicode[STIX]{x1D6FD}(r)=O(r^{4})$ and 
$\unicode[STIX]{x1D6FE}(r)=O(r^{2})$. Since 
$\unicode[STIX]{x1D711}_{1}=\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}=0$, we get that
$$\begin{eqnarray}(n-1)\unicode[STIX]{x1D711}_{2}=-\left.\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}}\right|_{r=0}-\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}-\frac{1}{2}\left.\left(\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\right)^{2}\left(\unicode[STIX]{x1D6FF}^{g,h}T\tilde{\unicode[STIX]{x1D711}}\right)^{(1)}\right|_{r=0}=0.\end{eqnarray}$$ By induction, we similarly have 
$\forall k\leqslant n,~\unicode[STIX]{x1D711}_{k}=0$, which implies that 
$P_{n}(\unicode[STIX]{x1D711})=0$.◻
Corollary 3.9. If 
$(M,H,J,\unicode[STIX]{x1D703})$ is Einstein and 
$(N,h)$ is a Kähler manifold, then CR-holomorphic maps which verify 
$R\unicode[STIX]{x1D711}=0$ are CR-harmonic.
Proof. Indeed, assuming that 
$T\unicode[STIX]{x1D711}\circ J=J_{N}\circ T\unicode[STIX]{x1D711}$ and extending 
$J$ by taking 
$J(R)=0$, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\unicode[STIX]{x1D711}^{\ast }h}T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711} & = & \displaystyle \unicode[STIX]{x1D6FB}_{JT_{\unicode[STIX]{x1D6FC}}}^{\unicode[STIX]{x1D711}^{\ast }h}JT_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & = & \displaystyle J_{N}\unicode[STIX]{x1D6FB}_{JT_{\unicode[STIX]{x1D6FC}}}^{\unicode[STIX]{x1D711}^{\ast }h}T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & = & \displaystyle J_{N}\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FC}}}}^{\unicode[STIX]{x1D711}^{\ast }h}JT_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711}+J([JT_{\unicode[STIX]{x1D6FC}},T_{\overline{\unicode[STIX]{x1D6FC}}}])\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FC}}}}^{\unicode[STIX]{x1D711}^{\ast }h}T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711}+iJ([T_{\unicode[STIX]{x1D6FC}},T_{\overline{\unicode[STIX]{x1D6FC}}}])\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & = & \displaystyle -\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FC}}}}^{\unicode[STIX]{x1D711}^{\ast }h}T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711}-nJ(R)\unicode[STIX]{x1D711};\nonumber\end{eqnarray}$$hence,
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}=nJ_{N}(R\unicode[STIX]{x1D711}).\end{eqnarray}$$ Consequently, 
$\unicode[STIX]{x1D711}$ is CR-harmonic by Corollary 3.8.◻
Example 3.10. Let 
$(M,H,J)$ be a circle bundle over a Riemann surface 
$\unicode[STIX]{x1D6F4}$ admitting an Einstein contact form. Then the projection 
$\unicode[STIX]{x1D70B}:M\rightarrow \unicode[STIX]{x1D6F4}$ is CR-harmonic.
3.4 Explicit obstruction in dimension 3
When 
$n=1$, that is, 
$\dim (M)=3$, the asymptotic development of 
$g$ is given at order 
$\frac{3}{2}$ in 
$e^{(0)}$ by Theorem 2.8. Hence, by Proposition 3.4, we can explicitly compute the obstruction.
Theorem 3.11. Still denoting 
$v\unicode[STIX]{x1D711}:=T\unicode[STIX]{x1D711}(v)$ for 
$v\in TM$, and also 
$(\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}v)\unicode[STIX]{x1D711}:=\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}(v\unicode[STIX]{x1D711})$, we have
$$\begin{eqnarray}P_{1}(\unicode[STIX]{x1D711})=-\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}-\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}+4\text{Im}\left(\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\unicode[STIX]{x1D711}^{\ast }h}\left(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}T_{\overline{1}}\right)\right)\unicode[STIX]{x1D711}-S_{b}\left(\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}\right),\end{eqnarray}$$where
$$\begin{eqnarray}S_{b}(X):={\mathcal{R}}_{X,T_{1}\unicode[STIX]{x1D711}}^{h}T_{\overline{1}}\unicode[STIX]{x1D711}+{\mathcal{R}}_{X,T_{\overline{1}}\unicode[STIX]{x1D711}}^{h}T_{1}\unicode[STIX]{x1D711}.\end{eqnarray}$$Proof. An orthonormal basis of 
$T^{1,0}X$ with respect to 
$g$ is given by
$$\begin{eqnarray}\displaystyle (e_{0},e_{1}):=\left(e_{0}^{(0)}-r^{3/2}\unicode[STIX]{x1D6F9}_{0\overline{1}}e_{1}^{(0)},\left(1-r\unicode[STIX]{x1D6F7}_{1\overline{1}}\right)e_{1}^{(0)}-r\unicode[STIX]{x1D6F7}_{11}e_{\overline{1}}^{(0)}\right)+O_{e}(r^{2}). & & \displaystyle \nonumber\end{eqnarray}$$We have
$$\begin{eqnarray}\displaystyle \,[e_{0},e_{\overline{0}}] & = & \displaystyle \frac{1}{\sqrt{2}}\left(e_{\overline{0}}-e_{0}-r^{3/2}\unicode[STIX]{x1D6F9}_{\overline{0}1}e_{\overline{1}}+r^{3/2}\unicode[STIX]{x1D6F9}_{0\overline{1}}e_{1}\right)+O_{e}(r^{2}),\nonumber\\ \displaystyle \,[e_{0},e_{1}] & = & \displaystyle \frac{1}{\sqrt{2}}\left(\left(\frac{1}{2}-r\unicode[STIX]{x1D6F7}_{1\overline{1}}\right)e_{1}-r\unicode[STIX]{x1D6F7}_{11}e_{\overline{1}}-i\big(\tilde{\unicode[STIX]{x1D6FB}}_{e_{0}}^{\unicode[STIX]{x1D703}}e_{1}-\unicode[STIX]{x1D70F}(e_{1})\big)\right)+O_{e}(r^{2}),\nonumber\\ \displaystyle \,[e_{0},e_{\overline{1}}] & = & \displaystyle \frac{1}{\sqrt{2}}\left(\left(\frac{1}{2}-r\unicode[STIX]{x1D6F7}_{1\overline{1}}\right)e_{\overline{1}}-r\unicode[STIX]{x1D6F7}_{\overline{1}\overline{1}}e_{1}-i\big(\tilde{\unicode[STIX]{x1D6FB}}_{e_{0}}^{\unicode[STIX]{x1D703}}e_{\overline{1}}-\unicode[STIX]{x1D70F}(e_{\overline{1}})\big)\right)+O_{e}(r^{2}),\nonumber\\ \displaystyle \,[e_{1},e_{\overline{1}}] & = & \displaystyle r^{3/2}\big(\unicode[STIX]{x1D6F7}_{1\overline{1},\overline{1}}-\unicode[STIX]{x1D6F7}_{\overline{1}\overline{1},1}\big)e_{1}-r^{3/2}\big(\unicode[STIX]{x1D6F7}_{1\overline{1},1}-\unicode[STIX]{x1D6F7}_{11,\overline{1}}\big)e_{\overline{1}}+O_{e}(r^{2}).\nonumber\end{eqnarray}$$Hence,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{e_{I}}^{g}e_{\overline{I}} & = & \displaystyle 2r\left(1-r\unicode[STIX]{x1D6F7}_{1\overline{1}}\right)\unicode[STIX]{x2202}_{r}\nonumber\\ \displaystyle & & \displaystyle -\,r^{2}\left(\sqrt{2}\unicode[STIX]{x1D6F9}_{0\overline{1}}+\unicode[STIX]{x1D6F7}_{1\overline{1},\overline{1}}-\unicode[STIX]{x1D6F7}_{\overline{1}\overline{1},1}\right)T_{1}\nonumber\\ \displaystyle & & \displaystyle -\,r^{2}\left(\sqrt{2}\unicode[STIX]{x1D6F9}_{\overline{0}1}+\unicode[STIX]{x1D6F7}_{1\overline{1},1}-\unicode[STIX]{x1D6F7}_{11,\overline{1}}\right)T_{\overline{1}}+O_{T}(r^{5/2}).\nonumber\end{eqnarray}$$ We also have, for 
$\unicode[STIX]{x1D714}\in \unicode[STIX]{x1D6FA}^{1}(X)\otimes \tilde{\unicode[STIX]{x1D711}}^{\ast }TN$,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{e_{0}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{0}})+\unicode[STIX]{x1D6FB}_{e_{\overline{0}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{0}) & = & \displaystyle r\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+r^{2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R)\nonumber\\ \displaystyle & & \displaystyle -\,\sqrt{2}r^{2}\unicode[STIX]{x1D6F9}_{0\overline{1}}\unicode[STIX]{x1D714}(T_{1})-\sqrt{2}r^{2}\unicode[STIX]{x1D6F9}_{\overline{0}1}\unicode[STIX]{x1D714}(T_{\overline{1}})+O_{T}(r^{5/2}),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FB}_{e_{1}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{1}}) & = & \displaystyle r\unicode[STIX]{x1D6FB}_{T_{1}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{1}})-r^{2}\unicode[STIX]{x1D6FB}_{T_{1}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\left(\unicode[STIX]{x1D6F7}_{1\overline{1}}\unicode[STIX]{x1D714}(T_{\overline{1}})\right)-r^{2}\unicode[STIX]{x1D6FB}_{T_{1}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\left(\unicode[STIX]{x1D6F7}_{\overline{1}\overline{1}}\unicode[STIX]{x1D714}(T_{1})\right)\nonumber\\ \displaystyle & & \displaystyle -\,r^{2}\unicode[STIX]{x1D6F7}_{1\overline{1}}\unicode[STIX]{x1D6FB}_{T_{1}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{1}})-r^{2}\unicode[STIX]{x1D6F7}_{11}\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{1}})+O_{T}(r^{3}).\nonumber\end{eqnarray}$$The divergence is hence given by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714} & = & \displaystyle r(1-2r\unicode[STIX]{x1D6F7}_{1\overline{1}})\left(\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-\unicode[STIX]{x1D6FB}_{T_{1}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{1}})-\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{1})\right)-r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-r^{2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R)\nonumber\\ \displaystyle & & \displaystyle +\,4r^{2}\text{Im}\left(\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\left(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}\unicode[STIX]{x1D714}(T_{\overline{1}})\right)\right)+O_{T}(r^{5/2})\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FF}^{g_{0},h}\unicode[STIX]{x1D714}-2r^{2}\unicode[STIX]{x1D6F7}_{1\overline{1}}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}+4r^{2}\text{Im}\left(\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\left(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}\unicode[STIX]{x1D714}(T_{\overline{1}})\right)\right)+O_{T}(r^{5/2}).\nonumber\end{eqnarray}$$Then, by Theorem 3.3, we have
$$\begin{eqnarray}\displaystyle P_{1}(\unicode[STIX]{x1D711}) & = & \displaystyle \unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\tilde{\unicode[STIX]{x1D6FF}}_{b}^{\unicode[STIX]{x1D703},h}T\tilde{\unicode[STIX]{x1D711}}|_{r=0}-\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}+4\text{Im}\left(\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\unicode[STIX]{x1D711}^{\ast }h}\left(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}T_{\overline{1}}\right)\right)\unicode[STIX]{x1D711}\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D711}_{1}+{\mathcal{R}}_{\unicode[STIX]{x1D711}_{1},T_{1}\unicode[STIX]{x1D711}}^{h}T_{\overline{1}}\unicode[STIX]{x1D711}+{\mathcal{R}}_{\unicode[STIX]{x1D711}_{1},T_{\overline{1}}\unicode[STIX]{x1D711}}^{h}T_{1}\unicode[STIX]{x1D711}-\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}+4\text{Im}\left(\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\unicode[STIX]{x1D711}^{\ast }h}\left(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}T_{\overline{1}}\right)\right)\unicode[STIX]{x1D711},\nonumber\end{eqnarray}$$ hence, since 
$\unicode[STIX]{x1D711}_{1}=-\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}$, the announced obstruction.◻
Note that on functions, meaning that 
$N=\mathbb{R}$, 
$P_{1}$ reduces to a multiple of the CR Paneitz operator. Since the construction follows the ideas of Graham et al., this was expected. A similar phenomenon appears in the real case [Reference BérardBér13].
Example 3.12. Let us consider 
$\text{id}:(M,H,J,\unicode[STIX]{x1D703})\rightarrow (M,g:=g_{J,\unicode[STIX]{x1D703}})$.
Since 
$\unicode[STIX]{x1D6FB}_{R}^{g}R=0$ by [Reference Dragomir and TomassiniDT06, Lemma 1.3], we have, using the Koszul formula,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},g}T\text{id} & = & \displaystyle -\unicode[STIX]{x1D6FB}_{T_{1}}^{g}T_{\overline{1}}-\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{g}T_{1}\nonumber\\ \displaystyle & = & \displaystyle -g\left([T_{1},T_{\overline{1}}],T_{1}\right)T_{\overline{1}}-g\left([T_{\overline{1}},T_{1}],T_{\overline{1}}\right)T_{1}\nonumber\\ \displaystyle & & \displaystyle -\,g\left([R,T_{\overline{1}}],T_{1}\right)R-g\left([R,T_{1}],T_{\overline{1}}\right)R\nonumber\\ \displaystyle & & \displaystyle -\,g\left([T_{1},R],R\right)T_{\overline{1}}-g\left([T_{\overline{1}},R],R\right)T_{1}\nonumber\\ \displaystyle & = & \displaystyle 0;\nonumber\end{eqnarray}$$hence,
$$\begin{eqnarray}P_{1}(\text{id})=4\text{Im}\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{g}\big(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}T_{\overline{1}}\big).\end{eqnarray}$$ Consequently, the identity is CR-harmonic if and only if 
$\text{Im}\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{g}(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}T_{\overline{1}})=0$. This is, in particular, verified when 
$\unicode[STIX]{x1D703}$ is normal, that is, when 
$\unicode[STIX]{x1D70F}=0$.
4 Renormalized energy
4.1 Definition
Let 
$\unicode[STIX]{x1D711}\in C^{\infty }(M,N)$ and 
$\tilde{\unicode[STIX]{x1D711}}$ be the extension of 
$\unicode[STIX]{x1D711}$ constructed in Theorem 3.3. For 
$\unicode[STIX]{x1D70C}$ in 
$(0,\unicode[STIX]{x1D700})$, let
$$\begin{eqnarray}E(\tilde{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D70C})=\frac{1}{2}\int _{(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D700})\times M}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}d\text{vol}_{g}\end{eqnarray}$$ be the energy of 
$\tilde{\unicode[STIX]{x1D711}}$ in 
$(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D700})\times M$. We have
$$\begin{eqnarray}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}=f_{0}r+f_{1}r^{2}+\cdots +f_{n}r^{n+1}+O(r^{n+2}\log r),\end{eqnarray}$$ where 
$\forall k\leqslant n,$
$f_{k}$ depends only on 
$U_{j}$ for 
$j\leqslant k$ and on 
$g$ at order 
$k$ in 
$e^{(0)}$, and
$$\begin{eqnarray}d\text{vol}_{g}=r^{-n-2}\sqrt{\det g}dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}.\end{eqnarray}$$Consequently,
$$\begin{eqnarray}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}d\text{vol}_{g}=\left(a_{0}r^{-n-1}+a_{1}r^{-n}+\cdots +a_{n}r^{-1}+O(\log r)\right)dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n},\end{eqnarray}$$ where 
$\forall k\leqslant n,$
$a_{k}$ depends only on 
$U_{j}$ for 
$j\leqslant k$ and on 
$g$ at order 
$k$. Hence, 
$E$ admits the development, when 
$\unicode[STIX]{x1D70C}\rightarrow 0$,
$$\begin{eqnarray}E(\tilde{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D70C})=E_{0}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{-n}+E_{1}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{1-n}+\cdots +E_{n-1}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{-1}+F_{n}(\unicode[STIX]{x1D711})\log \unicode[STIX]{x1D70C}+E_{n}(\unicode[STIX]{x1D711})+o(1),\end{eqnarray}$$ where 
$\forall k\leqslant n-1,$
$E_{k}$ depends only on 
$U_{j}$ for 
$j\leqslant k$ and on 
$g$ at order 
$k$ and 
$F_{n}$ depends only on 
$U_{j}$ for 
$j\leqslant n$ and on 
$g$ at order 
$n$. The coefficient 
$F_{n}(\unicode[STIX]{x1D711})$ can be written as
$$\begin{eqnarray}F_{n}(\unicode[STIX]{x1D711})=-\frac{1}{2}\int _{M}a_{n}\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}=-\frac{1}{2n!}\int _{M}\left.\unicode[STIX]{x2202}_{r}^{n}\left(r^{n+1}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}d\text{vol}_{g}\right)\right|_{r=0}.\end{eqnarray}$$ By construction, 
$F_{n}$ is formally determined by 
$\unicode[STIX]{x1D711}$, 
$g$ and 
$h$. Moreover, we have the following.
Proposition 4.1. 
$F_{n}(\unicode[STIX]{x1D711})$ is a CR invariant:
$$\begin{eqnarray}\hat{F}_{n}(\unicode[STIX]{x1D711})=F_{n}(\unicode[STIX]{x1D711}).\end{eqnarray}$$Proof. The proof is similar to the proof of Proposition 3.5. Indeed, if 
$\hat{r}=e^{f}r$, then
$$\begin{eqnarray}\displaystyle \Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}d\text{vol}_{g} & = & \displaystyle \big(a_{0}r^{-n-1}+a_{1}r^{-n}+\cdots +a_{n}r^{-1}+a_{n+1}+O(r)\big)dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}\nonumber\\ \displaystyle & = & \displaystyle \left(\hat{a}_{0}\hat{r}^{-n-1}+\hat{a}_{1}\hat{r}^{-n}+\cdots +\hat{a}_{n}\hat{r}^{-1}+\hat{a}_{n+1}+O(\hat{r})\right)d\hat{r}\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n};\nonumber\end{eqnarray}$$ hence, when integrating over 
$(r=\unicode[STIX]{x1D70C},r=\unicode[STIX]{x1D700})\times M$,
$$\begin{eqnarray}\displaystyle E(\tilde{\unicode[STIX]{x1D711}},\unicode[STIX]{x1D70C}) & = & \displaystyle E_{0}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{-n}+E_{1}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{1-n}+\cdots +E_{n-1}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{-1}+F_{n}(\unicode[STIX]{x1D711})\log \unicode[STIX]{x1D70C}+E_{n}(\unicode[STIX]{x1D711})+o(1)\nonumber\\ \displaystyle & = & \displaystyle \hat{E}_{0}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{-n}+\hat{E}_{1}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{1-n}+\cdots +\hat{E}_{n-1}(\unicode[STIX]{x1D711})\unicode[STIX]{x1D70C}^{-1}+\hat{F}_{n}(\unicode[STIX]{x1D711})\log \unicode[STIX]{x1D70C}+\hat{E}_{n}(\unicode[STIX]{x1D711})+o(1).\nonumber\end{eqnarray}$$ Again, since the function 
$f$ itself has a Taylor expansion in 
$r$, all polynomial terms are mixed. However, the only 
$\log \unicode[STIX]{x1D70C}$ term which appears when integrating with respect to 
$\hat{r}$ comes from the 
$\hat{r}^{-1}$ term. Hence, we have the result.◻
The principal term of 
$F_{n}(\unicode[STIX]{x1D711})$ is the following: since
$$\begin{eqnarray}\displaystyle r^{n+1}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2}d\text{vol}_{g} & = & \displaystyle \left(\left\langle T_{A}\tilde{\unicode[STIX]{x1D711}},T_{\overline{A}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}+r\Vert \unicode[STIX]{x2202}_{r}\tilde{\unicode[STIX]{x1D711}}\Vert _{h}^{2}\right)dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}\nonumber\\ \displaystyle & & \displaystyle +\,\text{lower order (in derivations of }\unicode[STIX]{x1D711}\text{) terms},\nonumber\end{eqnarray}$$we have
$$\begin{eqnarray}\displaystyle F_{n}(\unicode[STIX]{x1D711}) & = & \displaystyle -\frac{1}{2n!}\int _{M}\left(\mathop{\sum }_{k=0}^{n}\binom{n}{k}\langle \unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D711}_{k},\unicode[STIX]{x1D711}_{n-k}\rangle _{h}+n\mathop{\sum }_{k=0}^{n-1}\binom{n-1}{k}\left\langle \unicode[STIX]{x1D711}_{k+1},\unicode[STIX]{x1D711}_{n-k}\right\rangle _{h}\right)\unicode[STIX]{x1D703}\nonumber\\ \displaystyle & & \displaystyle \quad \wedge d\unicode[STIX]{x1D703}^{n}+\text{l.o.t.}\nonumber\\ \displaystyle & = & \displaystyle \frac{(-1)^{n+1}}{2n!^{2}}\int _{M}\langle (\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h})^{n-1}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711},\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}\rangle _{h}\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}+\text{lower order terms}.\nonumber\end{eqnarray}$$Definition 4.2. 
$F_{n}(\unicode[STIX]{x1D711})$ is called the renormalized energy of 
$\unicode[STIX]{x1D711}$.
Proposition 4.3. The gradient of 
$F_{n}(\unicode[STIX]{x1D711})$ is 
$(1/2n!)P_{n}(\unicode[STIX]{x1D711})$, that is to say, for all 
$\dot{\unicode[STIX]{x1D711}}\in \unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}^{\ast }TN)$,
$$\begin{eqnarray}d_{\unicode[STIX]{x1D711}}F_{n}(\dot{\unicode[STIX]{x1D711}})=\frac{1}{2n!}\int _{M}\left\langle \dot{\unicode[STIX]{x1D711}},P_{n}(\unicode[STIX]{x1D711})\right\rangle _{h}\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}.\end{eqnarray}$$Proof. Let 
$\dot{\unicode[STIX]{x1D711}}\in \unicode[STIX]{x1D6E4}(\unicode[STIX]{x1D711}^{\ast }TN)$. Let 
$(\unicode[STIX]{x1D711}_{t})_{t\in [-1,1]}$ be a one-parameter family in 
$C^{\infty }(M,N)$ such that
$$\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}r@{}}\unicode[STIX]{x1D711}_{0}=\unicode[STIX]{x1D711},\\ \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D711}_{t}|_{t=0}=\dot{\unicode[STIX]{x1D711}}.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$ Let us equip 
$X\times [-1,1]$ with the metric 
$\overline{g}=g+dt^{2}$ and let 
$\unicode[STIX]{x1D709}\in C^{\infty }(X\times [-1,1],N)$ be the map
$$\begin{eqnarray}\forall p\in X,~\forall t\in [-1,1],\quad \unicode[STIX]{x1D709}(p,t)=\tilde{\unicode[STIX]{x1D711}}_{t}(p).\end{eqnarray}$$We then have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}\Vert T\tilde{\unicode[STIX]{x1D711}}_{t}\Vert _{g,h}^{2} & = & \displaystyle \unicode[STIX]{x2202}_{t}\left(\Vert T\unicode[STIX]{x1D709}\Vert _{\overline{g},h}^{2}-\Vert \unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D709}\Vert _{h}^{2}\right)\nonumber\\ \displaystyle & = & \displaystyle \left\langle \unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{t}}^{\overline{g},h}T\unicode[STIX]{x1D709},T\unicode[STIX]{x1D709}\right\rangle _{\overline{g},h}-\left\langle \unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{t}}^{\unicode[STIX]{x1D709}^{\ast }h}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D709},\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D709}\right\rangle _{h}\nonumber\\ \displaystyle & = & \displaystyle \left\langle \unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{t}}^{\unicode[STIX]{x1D709}^{\ast }h}e_{I}\unicode[STIX]{x1D709},e_{\overline{I}}\unicode[STIX]{x1D709}\right\rangle _{h}\nonumber\\ \displaystyle & = & \displaystyle \left\langle \unicode[STIX]{x1D6FB}_{e_{I}}^{\unicode[STIX]{x1D709}^{\ast }h}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D709},e_{\overline{I}}\unicode[STIX]{x1D709}\right\rangle _{h}\nonumber\\ \displaystyle & = & \displaystyle \left\langle \unicode[STIX]{x1D6FB}_{e_{I}}^{\tilde{\unicode[STIX]{x1D711}}_{t}^{\ast }h}\unicode[STIX]{x2202}_{t}\tilde{\unicode[STIX]{x1D711}}_{t},e_{\overline{I}}\tilde{\unicode[STIX]{x1D711}}_{t}\right\rangle _{h}\nonumber\\ \displaystyle & = & \displaystyle e_{I}\left\langle \unicode[STIX]{x2202}_{t}\tilde{\unicode[STIX]{x1D711}}_{t},e_{\overline{I}}\tilde{\unicode[STIX]{x1D711}}_{t}\right\rangle _{h}-\left\langle \unicode[STIX]{x2202}_{t}\tilde{\unicode[STIX]{x1D711}}_{t},\unicode[STIX]{x1D6FB}_{e_{I}}^{\tilde{\unicode[STIX]{x1D711}}_{t}^{\ast }h}e_{\overline{I}}\tilde{\unicode[STIX]{x1D711}}_{t}\right\rangle _{h};\nonumber\end{eqnarray}$$hence,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}E(\tilde{\unicode[STIX]{x1D711}}_{t},\unicode[STIX]{x1D70C})|_{t=0}=\frac{1}{2}\int _{(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D700})\times M}\left(e_{I}\left\langle \unicode[STIX]{x2202}_{t}\tilde{\unicode[STIX]{x1D711}}_{t}|_{t=0},e_{\overline{I}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}-\left\langle \unicode[STIX]{x2202}_{t}\tilde{\unicode[STIX]{x1D711}}_{t}|_{t=0},\unicode[STIX]{x1D6FB}_{e_{I}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}e_{\overline{I}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}\right)d\text{vol}_{g}. & & \displaystyle \nonumber\end{eqnarray}$$ There is no 
$\log \unicode[STIX]{x1D70C}$ term in the second part, and
$$\begin{eqnarray}\displaystyle \frac{1}{2}\int _{(\unicode[STIX]{x1D70C},\unicode[STIX]{x1D700})\times M}e_{I}\left\langle \unicode[STIX]{x2202}_{t}\tilde{\unicode[STIX]{x1D711}}_{t}|_{t=0},e_{\overline{I}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}d\text{vol}_{g} & = & \displaystyle \frac{1}{2}\int _{M}\unicode[STIX]{x1D70C}^{-n}\left\langle \unicode[STIX]{x2202}_{t}\tilde{\unicode[STIX]{x1D711}}_{t}|_{t=0},\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70C}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}\nonumber\\ \displaystyle & & \displaystyle +\,\text{lower order terms},\nonumber\end{eqnarray}$$ whose 
$\log \unicode[STIX]{x1D70C}$ term is
$$\begin{eqnarray}\frac{1}{2n!}\int _{M}\left\langle \dot{\unicode[STIX]{x1D711}},P_{n}(\unicode[STIX]{x1D711})\right\rangle _{h}\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n};\end{eqnarray}$$hence, we have the result. ◻
4.2 Explicit energy in dimension 3
Here again, when 
$n=1$, that is, 
$\dim (M)=3$, knowing the asymptotic development of 
$g$ at order 
$\frac{3}{2}$ in 
$e^{(0)}$ allows for an explicit computation of the renormalized energy.
Theorem 4.4. We have
$$\begin{eqnarray}F_{1}(\unicode[STIX]{x1D711})=-\frac{1}{2}\int _{M}\big(\Vert \unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}\Vert _{h}^{2}+\Vert R\unicode[STIX]{x1D711}\Vert _{h}^{2}-4\text{Im}\big(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}\Vert T_{\overline{1}}\unicode[STIX]{x1D711}\Vert _{h}^{2}\big)\big)\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}.\end{eqnarray}$$Proof. We have
$$\begin{eqnarray}\displaystyle \Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2} & = & \displaystyle 2\left\langle e_{0}\tilde{\unicode[STIX]{x1D711}},e_{\overline{0}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}+2\left\langle e_{1}\tilde{\unicode[STIX]{x1D711}},e_{\overline{1}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}\nonumber\\ \displaystyle & = & \displaystyle 2r\left\langle T_{1}\unicode[STIX]{x1D711},T_{\overline{1}}\unicode[STIX]{x1D711}\right\rangle _{h}\nonumber\\ \displaystyle & & \displaystyle +\,r^{2}\left(\Vert \unicode[STIX]{x1D711}_{1}\Vert _{h}^{2}+\Vert R\unicode[STIX]{x1D711}\Vert _{h}^{2}-4\unicode[STIX]{x1D6F7}_{1\overline{1}}\left\langle T_{1}\unicode[STIX]{x1D711},T_{\overline{1}}\unicode[STIX]{x1D711}\right\rangle _{h}-2\unicode[STIX]{x1D6F7}_{11}\Vert T_{\overline{1}}\unicode[STIX]{x1D711}\Vert _{h}^{2}-2\unicode[STIX]{x1D6F7}_{\overline{1}\overline{1}}\Vert T_{1}\unicode[STIX]{x1D711}\Vert _{h}^{2}\right)\nonumber\\ \displaystyle & & \displaystyle +\,O(r^{5/2}),\nonumber\end{eqnarray}$$and
$$\begin{eqnarray}d\text{vol}_{g}=\left(1+2r\unicode[STIX]{x1D6F7}_{1\overline{1}}+O(r^{2})\right)r^{-3}dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}.\end{eqnarray}$$Consequently,
$$\begin{eqnarray}\displaystyle r^{2}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g}^{2}d\text{vol}_{g} & = & \displaystyle \big(2\langle T_{1}\unicode[STIX]{x1D711},T_{\overline{1}}\unicode[STIX]{x1D711}\rangle _{h}\nonumber\\ \displaystyle & & \displaystyle +\,r\big(\Vert \unicode[STIX]{x1D711}_{1}\Vert _{h}^{2}+\Vert R\unicode[STIX]{x1D711}\Vert _{h}^{2}-2\unicode[STIX]{x1D6F7}_{11}\Vert T_{\overline{1}}\unicode[STIX]{x1D711}\Vert _{h}^{2}-2\unicode[STIX]{x1D6F7}_{\overline{1}\overline{1}}\Vert T_{1}\unicode[STIX]{x1D711}\Vert _{h}^{2}\big)\nonumber\\ \displaystyle & & \displaystyle +\,O(r^{3/2})\big)dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703},\nonumber\end{eqnarray}$$and finally,
$$\begin{eqnarray}F_{1}(\unicode[STIX]{x1D711})=-\frac{1}{2}\int _{M}\big(\Vert \unicode[STIX]{x1D711}_{1}\Vert _{h}^{2}+\Vert R\unicode[STIX]{x1D711}\Vert _{h}^{2}-4\text{Im}\big(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}\Vert T_{\overline{1}}\unicode[STIX]{x1D711}\Vert _{h}^{2}\big)\big)\unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}.\square\end{eqnarray}$$ As an example, for 
$\text{id}:(M,H,J,\unicode[STIX]{x1D703})\rightarrow (M,g_{J,\unicode[STIX]{x1D703}})$, we have
$$\begin{eqnarray}F_{1}(\text{id})=-{\textstyle \frac{1}{2}}\text{Vol}(M,\unicode[STIX]{x1D703}).\end{eqnarray}$$5 Further computations in the general case
We give here a more precise computation for 
$\unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714}$ and 
$r^{n+1}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g}^{2}d\text{vol}_{g}$ in the general case, using Theorem 2.4. We show that this computation does not allow for an explicit expression of the obstruction and of the renormalized energy respectively.
5.1 Computation of the divergence
By Theorem 2.4, we have
$$\begin{eqnarray}g=g_{0}+\unicode[STIX]{x1D6F7}_{AB}\unicode[STIX]{x1D703}^{A}\circ \unicode[STIX]{x1D703}^{B}+O_{e}(r^{3/2}),\end{eqnarray}$$ where, denoting by 
$R_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}$ the components of the Webster Ricci tensor,
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}=-\frac{1}{n+2}\left(R_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}-\frac{\text{Scal}_{W}(J,\unicode[STIX]{x1D703})}{2(n+1)}\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}\right),\qquad \text{and}\qquad \unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}=-i\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}^{\overline{\unicode[STIX]{x1D6FD}}}.\end{eqnarray}$$ By Proposition 2.6, we can equip 
$\{r\}\times H$ with a complex structure 
$J_{r}=J_{0}+rJ_{1}+O_{T}(r^{2})$, with
$$\begin{eqnarray}J_{1}T_{\unicode[STIX]{x1D6FC}}=-2\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}T_{\overline{\unicode[STIX]{x1D6FD}}}.\end{eqnarray}$$ An orthonormal basis of 
$T^{1,0}X$ with respect to 
$g$ is given by
$$\begin{eqnarray}\displaystyle (e_{0},e_{\unicode[STIX]{x1D6FC}}):=\left(r\unicode[STIX]{x2202}_{r}-irR,\left(\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}-r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}\right)r^{1/2}T_{\unicode[STIX]{x1D6FD}}-r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}r^{1/2}T_{\overline{\unicode[STIX]{x1D6FD}}}\right)+O_{e}(r^{3/2}). & & \displaystyle \nonumber\end{eqnarray}$$ Now, 
$g$ can be rewritten as
$$\begin{eqnarray}g=(r^{-1}\unicode[STIX]{x1D703}^{0})\circ (r^{-1}\unicode[STIX]{x1D703}^{\overline{0}})+(r^{-1/2}\unicode[STIX]{x1D703}^{\unicode[STIX]{x1D6FC}})\circ (r^{-1/2}\unicode[STIX]{x1D703}^{\overline{\unicode[STIX]{x1D6FC}}})+r\unicode[STIX]{x1D6F7}_{AB}(r^{-1/2}\unicode[STIX]{x1D703}^{A})\circ (r^{-1/2}\unicode[STIX]{x1D703}^{B})+O_{e}(r^{3/2}).\end{eqnarray}$$ We have, modulo 
$O_{e}(r^{3/2})$,
$$\begin{eqnarray}\displaystyle \,[e_{0},e_{\overline{0}}] & = & \displaystyle \frac{1}{\sqrt{2}}\left(e_{\overline{0}}-e_{0}\right),\nonumber\\ \displaystyle \,[e_{0},e_{\unicode[STIX]{x1D6FC}}] & = & \displaystyle \frac{1}{\sqrt{2}}\left(\frac{1}{2}e_{\unicode[STIX]{x1D6FC}}-r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}e_{\unicode[STIX]{x1D6FD}}-r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}e_{\overline{\unicode[STIX]{x1D6FD}}}-i\big(\tilde{\unicode[STIX]{x1D6FB}}_{e_{0}}^{\unicode[STIX]{x1D703}}e_{\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D70F}(e_{\unicode[STIX]{x1D6FC}})\big)\right),\nonumber\\ \displaystyle \,[e_{0},e_{\overline{\unicode[STIX]{x1D6FC}}}] & = & \displaystyle \frac{1}{\sqrt{2}}\left(\frac{1}{2}e_{\overline{\unicode[STIX]{x1D6FC}}}-r\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FD}}e_{\overline{\unicode[STIX]{x1D6FD}}}-r\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}}}e_{\unicode[STIX]{x1D6FD}}-i\big(\tilde{\unicode[STIX]{x1D6FB}}_{e_{0}}^{\unicode[STIX]{x1D703}}e_{\overline{\unicode[STIX]{x1D6FC}}}-\unicode[STIX]{x1D70F}(e_{\overline{\unicode[STIX]{x1D6FC}}})\big)\right),\nonumber\\ \displaystyle \,[e_{\unicode[STIX]{x1D6FC}},e_{\unicode[STIX]{x1D6FD}}] & = & \displaystyle r^{3/2}\big(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FD}\overline{\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D6FC}}\big)e_{\unicode[STIX]{x1D6FF}}+r^{3/2}\left(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FC}}\right)e_{\overline{\unicode[STIX]{x1D6FF}}},\nonumber\\ \displaystyle \,[e_{\unicode[STIX]{x1D6FC}},e_{\overline{\unicode[STIX]{x1D6FD}}}] & = & \displaystyle r^{3/2}\big(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FF}},\overline{\unicode[STIX]{x1D6FD}}}-\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FD}}\overline{\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D6FC}}\big)e_{\unicode[STIX]{x1D6FF}}-r^{3/2}\big(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FF}\overline{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF},\overline{\unicode[STIX]{x1D6FD}}}\big)e_{\overline{\unicode[STIX]{x1D6FF}}}.\nonumber\end{eqnarray}$$Hence,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{e_{i}}^{g}e_{\overline{i}}+\unicode[STIX]{x1D6FB}_{e_{\overline{i}}}^{g}e_{i} & = & \displaystyle r\left(n+1-2r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FC}}}\right)\unicode[STIX]{x2202}_{r}\nonumber\\ \displaystyle & & \displaystyle -\,r^{2}\big(2\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FD}\overline{\unicode[STIX]{x1D6FD}},\overline{\unicode[STIX]{x1D6FC}}}-\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FD}}-\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FD},\overline{\unicode[STIX]{x1D6FD}}}\big)T_{\unicode[STIX]{x1D6FC}}\nonumber\\ \displaystyle & & \displaystyle -\,r^{2}\big(2\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FD}\overline{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FC}}-\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD},\overline{\unicode[STIX]{x1D6FD}}}-\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}},\unicode[STIX]{x1D6FD}}\big)T_{\overline{\unicode[STIX]{x1D6FC}}}+O_{T}(r^{5/2}),\nonumber\end{eqnarray}$$ with 
$\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FC}}}=-\text{tr}(\text{S}_{\unicode[STIX]{x1D703}})=-\text{Scal}_{W}/2(n+1)$.
Also,
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{e_{0}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{0}})+\unicode[STIX]{x1D6FB}_{e_{\overline{0}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{0}) & = & \displaystyle r\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})+r^{2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(R)+O_{T}(r^{2}),\nonumber\\ \displaystyle \unicode[STIX]{x1D6FB}_{e_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(e_{\overline{\unicode[STIX]{x1D6FC}}}) & = & \displaystyle r\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})-r^{2}\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big(\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FD}}})\big)-r^{2}\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big(\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}(T_{\unicode[STIX]{x1D6FD}})\big)\nonumber\\ \displaystyle & & \displaystyle -\,r^{2}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FD}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})-r^{2}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FD}}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})+O_{T}(r^{5/2}).\nonumber\end{eqnarray}$$Coming back to the divergence, we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}^{g,h}\unicode[STIX]{x1D714} & = & \displaystyle r(n-2r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FC}}})\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-r^{2}\unicode[STIX]{x1D6FB}_{\unicode[STIX]{x2202}_{r}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(\unicode[STIX]{x2202}_{r})-r^{2}\unicode[STIX]{x1D6FB}_{R}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}(\unicode[STIX]{x1D714}(R))\nonumber\\ \displaystyle & & \displaystyle -\,r(1-2r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FC}}})\big(\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})+\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FC}}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D714}(T_{\unicode[STIX]{x1D6FC}})\big)\nonumber\\ \displaystyle & & \displaystyle +\,2r^{2}\big(\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FD}}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})\big)+\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FD}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big(\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}(T_{\unicode[STIX]{x1D6FC}})\big)\big)\nonumber\\ \displaystyle & & \displaystyle +\,2r^{2}\big(\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FD}}})+\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FC}}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}(T_{\unicode[STIX]{x1D6FD}})-\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FC}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FD}\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})-\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FC}}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FD}\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}(T_{\unicode[STIX]{x1D6FC}})\big)\nonumber\\ \displaystyle & & \displaystyle +\,O_{T}(r^{2}).\nonumber\end{eqnarray}$$ The term of order 
$2$ is consequently not known, which does not allow for an explicit computation of 
$P_{n}$. Note that
$$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FD}}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\left(\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})\right)+\unicode[STIX]{x1D6FB}_{T_{\unicode[STIX]{x1D6FD}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big(\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}(T_{\unicode[STIX]{x1D6FC}})\big)=2\text{Im}\big(\unicode[STIX]{x1D6FB}_{T_{\overline{\unicode[STIX]{x1D6FD}}}}^{\tilde{\unicode[STIX]{x1D711}}^{\ast }h}\big(\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FC}}^{\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})\big)\big)\end{eqnarray}$$ and that the potentially hidden 
$r^{2}$ terms are necessarily of the form 
$C^{\unicode[STIX]{x1D6FC}}r^{2}\unicode[STIX]{x1D714}(T_{\unicode[STIX]{x1D6FC}})+D^{\overline{\unicode[STIX]{x1D6FC}}}r^{2}\unicode[STIX]{x1D714}(T_{\overline{\unicode[STIX]{x1D6FC}}})$.
5.2 Computation of the integrand of the energy
We have
$$\begin{eqnarray}\displaystyle \Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g,h}^{2} & = & \displaystyle 2\left\langle e_{0}\tilde{\unicode[STIX]{x1D711}},e_{\overline{0}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}+2\left\langle e_{\unicode[STIX]{x1D6FC}}\tilde{\unicode[STIX]{x1D711}},e_{\overline{\unicode[STIX]{x1D6FC}}}\tilde{\unicode[STIX]{x1D711}}\right\rangle _{h}\nonumber\\ \displaystyle & = & \displaystyle 2r\left\langle T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711},T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711}\right\rangle _{h}\nonumber\\ \displaystyle & & \displaystyle +\,r^{2}\left(\Vert \unicode[STIX]{x1D711}_{1}\Vert _{h}^{2}+\Vert R\unicode[STIX]{x1D711}\Vert _{h}^{2}-2\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\langle T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711},T_{\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D711}\rangle _{h}-2\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}}}\left\langle T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711},T_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D711}\right\rangle _{h}\right.\nonumber\\ \displaystyle & & \displaystyle \left.-2\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}\left\langle T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711},T_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D711}\right\rangle _{h}-2\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FD}}\langle T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711},T_{\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D711}\rangle _{h}\right)\nonumber\\ \displaystyle & & \displaystyle +\,O(r^{2}),\nonumber\end{eqnarray}$$and
$$\begin{eqnarray}d\text{vol}_{g}=\big(1+2r\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FC}}}+O(r^{3/2})\big)r^{-n-2}dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}.\end{eqnarray}$$Consequently,
$$\begin{eqnarray}\displaystyle r^{n+1}\Vert T\tilde{\unicode[STIX]{x1D711}}\Vert _{g}^{2}d\text{vol}_{g} & = & \displaystyle \big(2\langle T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711},T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711}\rangle _{h}\nonumber\\ \displaystyle & & \displaystyle +\,r\big(\Vert \unicode[STIX]{x1D711}_{1}\Vert _{h}^{2}+\Vert R\unicode[STIX]{x1D711}\Vert _{h}^{2}-2\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\langle T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711},T_{\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D711}\rangle _{h}-2\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\overline{\unicode[STIX]{x1D6FD}}}\langle T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711},T_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D711}\rangle _{h}\nonumber\\ \displaystyle & & \displaystyle -\,2\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FD}}}\langle T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711},T_{\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D711}\rangle _{h}-2\unicode[STIX]{x1D6F7}_{\overline{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D6FD}}\langle T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711},T_{\overline{\unicode[STIX]{x1D6FD}}}\unicode[STIX]{x1D711}\rangle _{h}+4\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D6FC}\overline{\unicode[STIX]{x1D6FC}}}\langle T_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D711},T_{\overline{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D711}\rangle _{h}\big)\nonumber\\ \displaystyle & & \displaystyle +\,O(r)\big)dr\wedge \unicode[STIX]{x1D703}\wedge d\unicode[STIX]{x1D703}^{n}.\nonumber\end{eqnarray}$$ The term of order 
$1$ is consequently not known, which does not allow for an explicit computation of 
$F_{n}$.
6 Relation with the Fefferman bundle in dimension 
$3$
We describe here the correspondence between the obstruction to CR-harmonicity on a given CR 
$3$-manifold and the obstruction to conformal harmonicity on its Fefferman bundle. It generalizes the Appendix B. of [Reference Case and YangCY13].
Let 
$(M,H,J)$ be a compact strictly pseudoconvex CR 
$3$-manifold and let 
$(N,h)$ be a Riemannian manifold. Let 
$(F,g_{F})$ be the Fefferman bundle of 
$(M,H,J)$. For a detailed construction of the Fefferman bundle, see [Reference FarrisFar86, Reference LeeLee86, Reference HerzlichHer09]. Let 
$\unicode[STIX]{x1D70B}:F\rightarrow M$ be the natural bundle projection. Let 
$\unicode[STIX]{x1D703}$ be a positive contact form on 
$M$ and let 
$\unicode[STIX]{x1D71B}$ be the 
$S^{1}$-invariant connection 
$1$-form induced by the Weyl structure attached to 
$\unicode[STIX]{x1D703}$ on 
$F$. The Fefferman metric attached to 
$\unicode[STIX]{x1D703}$ on 
$F$ is the Lorentzian metric
$$\begin{eqnarray}g_{F}=i\unicode[STIX]{x1D71B}\circ \unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D703}+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D6FE}.\end{eqnarray}$$ By analogy with the Riemannian case [Reference BérardBér13], given 
$\unicode[STIX]{x1D711}\in C^{\infty }(F,N)$, the obstruction to the existence of a smooth harmonic extension of 
$\unicode[STIX]{x1D711}$ on the interior of 
$(F,g_{F})$ is given by
$$\begin{eqnarray}P_{F}(\unicode[STIX]{x1D711})=-{\textstyle \frac{1}{16}}\big(\unicode[STIX]{x1D6FF}^{g_{F},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D6FF}^{g_{F},h}T\unicode[STIX]{x1D711}-\unicode[STIX]{x1D6FF}^{g_{F},h}\big(2\text{Ric}_{g_{F}}-{\textstyle \frac{2}{3}}\text{Scal}_{g_{F}}\big)T\unicode[STIX]{x1D711}+S(\unicode[STIX]{x1D6FF}^{g_{F},h}T\unicode[STIX]{x1D711})\big),\end{eqnarray}$$ where 
$\text{Ric}_{g_{F}}$ is understood as an endomorphism of 
$TF$, and 
$\text{Ric}_{g_{F}}T\unicode[STIX]{x1D711}:=T\unicode[STIX]{x1D711}(\text{Ric}_{g_{F}}(\cdot ))$, and
$$\begin{eqnarray}S(X):=\mathop{\sum }_{i=1}^{4}{\mathcal{R}}_{X,T\unicode[STIX]{x1D711}(e_{i})}^{h}T\unicode[STIX]{x1D711}(e_{i}).\end{eqnarray}$$Proposition 6.1. For all 
$\unicode[STIX]{x1D711}\in C^{\infty }(M^{3},N)$,
$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70B}_{\ast }\left(\unicode[STIX]{x1D6FF}^{g_{F},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D6FF}^{g_{F},h}T(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D711})\right)=4\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}\unicode[STIX]{x1D6FB}^{\unicode[STIX]{x1D711}^{\ast }h}\unicode[STIX]{x1D6FF}_{b}^{\unicode[STIX]{x1D703},h}T\unicode[STIX]{x1D711}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D70B}_{\ast }\left(\unicode[STIX]{x1D6FF}^{g_{F},h}\left(2\text{Ric}_{g_{F}}-{\textstyle \frac{2}{3}}\text{Scal}_{g_{F}}\right)T(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D711})\right)=-4\unicode[STIX]{x1D6FB}_{R}^{\unicode[STIX]{x1D711}^{\ast }h}R\unicode[STIX]{x1D711}+16\text{Im}\big(\unicode[STIX]{x1D6FB}_{T_{\overline{1}}}^{\unicode[STIX]{x1D711}^{\ast }h}\big(\unicode[STIX]{x1D70F}_{1}^{\overline{1}}T_{\overline{1}}\big)\big)\unicode[STIX]{x1D711}, & \displaystyle \nonumber\end{eqnarray}$$ and for 
$X$ in 
$TN$,
$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }\left(S((\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D711})^{\ast }X)\right)=4S_{b}(\unicode[STIX]{x1D711}^{\ast }X).\end{eqnarray}$$Proof. The first and third equalities are straightforward from the expression of 
$g_{F}$. The second equality comes from the fact that, see [Reference LeeLee86],
$$\begin{eqnarray}\text{Sch}_{g_{F}}=-\unicode[STIX]{x1D71B}^{2}-\mathbf{S}\unicode[STIX]{x1D703}^{2}+{\textstyle \frac{1}{2}}\text{Sch}_{W}-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FE}(J\unicode[STIX]{x1D70F}\cdot ,\cdot )+{\textstyle \frac{1}{2}}\mathbf{T}J\circ \unicode[STIX]{x1D703},\end{eqnarray}$$where
$$\begin{eqnarray}\mathbf{T}={\textstyle \frac{1}{3}}\left({\textstyle \frac{1}{4}}d_{b}\text{Scal}_{W}+i\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70F}\right)\qquad \text{and}\qquad \mathbf{S}=\unicode[STIX]{x1D6FF}\mathbf{T}-|\text{Sch}_{W}|^{2}+|\unicode[STIX]{x1D70F}|^{2}.\end{eqnarray}$$ Indeed, since 
$\text{Scal}_{g_{F}}=3\text{Scal}_{W}~$ and 
$~\text{Sch}_{W}=\frac{1}{4}\text{Scal}_{W}\unicode[STIX]{x1D6FE}$, we have then
$$\begin{eqnarray}\displaystyle 2\text{Ric}_{g_{F}}-{\textstyle \frac{2}{3}}\text{Scal}_{g_{F}}g_{F} & = & \displaystyle 4\text{Sch}_{g_{F}}-{\textstyle \frac{1}{3}}\text{Scal}_{g_{F}}g_{F}\nonumber\\ \displaystyle & = & \displaystyle -4\unicode[STIX]{x1D71B}^{2}-4\mathbf{S}\unicode[STIX]{x1D703}^{2}-2\unicode[STIX]{x1D6FE}(J\unicode[STIX]{x1D70F}\cdot ,\cdot )+2\mathbf{T}J\circ \unicode[STIX]{x1D703}-\text{Scal}_{W}i\unicode[STIX]{x1D71B}\circ \unicode[STIX]{x1D703},\nonumber\end{eqnarray}$$which gives the second equality. ◻
From the latter comes directly the following.
Theorem 6.2. For all 
$\unicode[STIX]{x1D711}\in C^{\infty }(M^{3},N)$,
$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }\left(P_{F}(\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D711})\right)={\textstyle \frac{1}{4}}P_{1}(\unicode[STIX]{x1D711}).\end{eqnarray}$$ In particular, a map 
$\unicode[STIX]{x1D711}:M\rightarrow N$ is CR-harmonic if and only if 
$\unicode[STIX]{x1D70B}^{\ast }\unicode[STIX]{x1D711}$ is conformal-harmonic.
Acknowledgments
I am deeply grateful to my supervisor, Marc Herzlich, for introducing me to these questions, for his numerous advices and his precious help. I also profusely thank the anonymous referee of this paper and the referees of my PhD thesis, Jih-Hsin Cheng and Colin Guillarmou, for their careful reading.
