On the Yamabe problem on contact Riemannian manifolds

Abstract

A contact Riemannian manifold, whose complex structure is not necessarily integrable, is the generalization of the notion of a pseudohermitian manifold in CR geometry. The Tanaka–Webster–Tanno connection plays the role of the Tanaka–Webster connection for a pseudohermitian manifold. Conformal transformations and the Yamabe problem are also defined naturally in this setting. By using special frames and normal coordinates on a contact Riemannian manifold, we prove that if the complex structure is not integrable, the Yamabe invariant on a contact Riemannian manifold is always less than the Yamabe invariant of the Heisenberg group. So the Yamabe problem on a contact Riemannian manifold is always solvable.

Appendices

Appendix A: Transformation formulae under a conformal transformation

In this appendix, we will discuss conformal transformations and prove Lemma 4.1. We write components of the tensors after conformal transformation with respect to $\{W_a,\widehat{T}\}$ and $\{\widehat{\theta }^a,\widehat{\theta }\}$ satisfying (4.1)–(4.3), e.g., $\tau (\widehat{T},W_a)=\widehat{A}_a^bW_b,$$\Gamma _{\widehat{0}b}^c=\omega _b^c(\widehat{T}).$ See [5, Lemma 10] for a version for a local $T^{(1,0)}M$-frame.

1.1 A.1 The transformation formula for connection coefficients

Lemma A.1

We have

$$\begin{aligned} 2h(\nabla _XY,Z)=&X(h(Y,Z))+Y(h(X,Z))-Z(h(X,Y))\nonumber \\&-2h(X,JZ)\theta (Y)-2h(Y,JZ)\theta (X)+2h(X,JY)\theta (Z)\nonumber \\&-h([X,Z],Y)-h([Y,Z],X)+h([X,Y],Z), \end{aligned}$$

(A.1)

for any $X,Y,Z\in {TM}$. And also we have

$$\begin{aligned} 2h(\nabla _TY,Z)=T(h(Y,Z))-h([T,Z],Y)+h([T,Y],Z), \end{aligned}$$

(A.2)

for any $Y,Z{\in }HM$.

Proof

We refer to [5, p. 334] for (A.1). For (A.2), we have

$$\begin{aligned} T(h(Y,Z))&=h(\nabla _TY,Z)+h(Y,\nabla _TZ)=h(\nabla _TY,Z)+h(Y,[T,Z])+h(Y,\tau _{*}Z)\\&=h(\nabla _TY,Z)+h(Y,[T,Z])+h(\tau _{*}Y,Z)\\&=2h(\nabla _TY,Z)+h([T,Z],Y)-h([T,Y],Z), \end{aligned}$$

by $\nabla {T}=0$, the definition of the Webster torsion $\tau _{*}$ and its self-adjointness (cf. Lemma 2.1). $\square$

Corollary A.1

With respect to a frame $\{W_a,T\}$ with $\{W_a\}$ horizontal, we have

$$\begin{aligned} \Gamma _{ab}^c=&\frac{1}{2}h^{cd}\Big (W_a(h_{bd})+W_b(h_{ad})-W_d(h_{ab})\nonumber \\&-h([W_a,W_d],W_b)-h([W_b,W_d],W_a)+h([W_a,W_b],W_d)\Big ), \end{aligned}$$

(A.3)

and

$$\begin{aligned} \Gamma _{0b}^c=\frac{1}{2}h^{cd}\Big (T h_{bd} -h([T,W_d],W_b)+h([T,W_b],W_d)\Big ). \end{aligned}$$

(A.4)

Proof

(A.3) follows from substituting $X=W_a$, $Y=W_b$, $Z=W_d$ into (A.1). (A.4) follows from substituting $Y=W_b$, $Z=W_d$ into (A.2). $\square$

Lemma A.2

Under the conformal transformation (4.1) with $u\in {\mathscr {O}_m}$, we have

$$\begin{aligned} {[}\widehat{T},W_{\beta }]=[T,W_{\beta }]-iZ_{\beta }Z_{\bar{\alpha }}uW_{\alpha }+iZ_{\beta }Z_{\alpha }uW_{\bar{\alpha }}+\mathscr {E}_{m-1}(W), \end{aligned}$$

(A.5)

where $\mathscr {E}_{m-1}(W)$ denotes the linear combination of $W_j$’s with coefficients in $\mathscr {O}_{m-1}$.

Proof

We have

$$\begin{aligned} {[}\widehat{T},W_{\beta }]&=[e^{-2u}(T+J_{\ a}^cu^aW_c),W_{\beta }]\\&=e^{-2u}[T,W_{\beta }]+e^{-2u}[J_{\ a}^cu^aW_c,W_{\beta }]+2e^{-2u}u_{\beta }(T+J_{\ a}^cu^aW_c)\\&=[T,W_{\beta }]-(W_{\beta }u^a)J_{\ a}^cW_c+\mathscr {O}_{m-1}\mathscr {E}(W)\\&=[T,W_{\beta }]-(W_{\beta }u_{\bar{\mu }})h^{\alpha \bar{\mu }}J_{\ \alpha }^{\rho }(q)W_{\rho }-(W_{\beta }u_{\mu })h^{\mu \bar{\alpha }}J_{\ \bar{\alpha }}^{\bar{\rho }}(q)W_{\bar{\rho }}+\mathscr {E}_{m-1}(W)\\&=[T,W_{\beta }]-iZ_{\beta }Z_{\bar{\alpha }}uW_{\alpha }+iZ_{\beta }Z_{\alpha }uW_{\bar{\alpha }}+\mathscr {E}_{m-1}(W), \end{aligned}$$

by $h_{\alpha \bar{\beta }}=\delta _{\alpha \bar{\beta }}$, $J_{\ \alpha }^{\rho }(q)=i\delta _{\ \alpha }^{\rho }$ in (2.14) and $u_a{\in }\mathscr {O}_{m-1}$ for $u\in \mathscr {O}_m$. (A.5) follows. $\square$

Proposition A.1

Under the conformal transformation (4.1), connection coefficients of TWT connections change as

$$\begin{aligned} \begin{aligned} \widehat{\Gamma }_{ab}^c&=\Gamma _{ab}^c+u_a\delta _b^c+u_b\delta _a^c-u^ch_{ab},\\ \widehat{\Gamma }_{\widehat{0}\beta }^{\rho }&=\Gamma _{0\beta }^{\rho }+u_0\delta _{\beta }^{\rho } -\frac{i}{2}(Z_{\bar{\rho }}Z_{\beta }u+Z_{\beta }Z_{\bar{\rho }}u)+\mathscr {O}_{m-1}. \end{aligned} \end{aligned}$$

(A.6)

Proof

By Corollary A.1, we get

$$\begin{aligned} \widehat{\Gamma }_{ab}^c&=\frac{1}{2}e^{-2u}h^{cd} \bigg (W_a\bigg (e^{2u}h_{bd}\bigg )+W_b\bigg (e^{2u}h_{ad}\bigg ) -W_d\bigg (e^{2u}h_{ab}\bigg )\\&\quad -e^{2u}h([W_a,W_d],W_b)-e^{2u}h([W_b,W_d],W_a) +e^{2u}h([W_a,W_b],W_d)\bigg )\\&=\Gamma _{ab}^c+u_a\delta _b^c+u_b\delta _a^c-u^ch_{ab}. \end{aligned}$$

Note that

$$\begin{aligned} \widehat{\Gamma }_{\widehat{0}\beta }^{\rho }=\frac{1}{2}\widehat{h}^{\rho \bar{\mu }}\left( \widehat{T} \widehat{h}_{\beta \bar{\mu }} -\widehat{h}\left( \left[ \widehat{T},W_{\bar{\mu }}\right] ,W_{\beta }\right) +\widehat{h}\left( \left[ \widehat{T},W_{\beta }\right] ,W_{\bar{\mu }}\right) \right) . \end{aligned}$$

(A.7)

For the first term in the right-hand side of (A.7), by (4.2), we have

$$\begin{aligned} \frac{1}{2}\widehat{h}^{\rho \bar{\mu }}\widehat{T} \widehat{h}_{\beta \bar{\mu }}&=\frac{1}{2}e^{-4u}h^{\rho \bar{\mu }}\left( T+J_{\ a}^eu^aW_e\right) \left( e^{2u}h_{\beta \bar{\mu }}\right) =\frac{1}{2}h^{\rho \bar{\mu }}T h_{\beta \bar{\mu }} +u_0\delta _{\beta }^{\rho }+\mathscr {O}_{m-1}. \end{aligned}$$

Take conjugation on both sides of (A.5) to get $[\widehat{T},W_{\bar{\mu }}]=[T,W_{\bar{\mu }}]+iZ_{\bar{\mu }}Z_{\alpha }uW_{\bar{\alpha }}- iZ_{\bar{\mu }}Z_{\bar{\alpha }}uW_{\alpha }+\mathscr {E}_{m-1}(W)$ and so

$$\begin{aligned} -\frac{1}{2}\widehat{h}^{\rho \bar{\mu }}\widehat{h}([\widehat{T},W_{\bar{\mu }}],W_{\beta }) =-\frac{1}{2}h^{\rho \bar{\mu }}h([T,W_{\bar{\mu }}],W_{\beta })-\frac{i}{2}Z_{\bar{\rho }}Z_{\beta }u+\mathscr {O}_{m-1}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2}\widehat{h}^{\rho \bar{\mu }}\widehat{h}([\widehat{T},W_{\beta }],W_{\bar{\mu }}) =\frac{1}{2}h^{\rho \bar{\mu }}h([T,W_{\beta }],W_{\bar{\mu }})-\frac{i}{2}Z_{\beta }Z_{\bar{\rho }}u+\mathscr {O}_{m-1}. \end{aligned}$$

So (A.7) becomes $\widehat{\Gamma }_{\widehat{0}\beta }^{\rho }=\Gamma _{0\beta }^{\rho }+u_0\delta _{\beta }^{\rho } -\frac{i}{2}(Z_{\bar{\rho }}Z_{\beta }u+Z_{\beta }Z_{\bar{\rho }}u)+\mathscr {O}_{m-1}$. $\square$

1.2 A.2. Transformation formulae for curvature and Webster torsion tensors

Proof of Lemma 4.1

By $\nabla {T}=0$ and $\tau _{*}W_a=\tau (T,W_a)=\nabla _TW_a-[T,W_a]$, we get

$$\begin{aligned} A_{ab}&=h(A_a^cW_c,W_b)=h(\tau _{*}W_a,W_b)=h(\nabla _TW_a-[T,W_a],W_b)\\&=T(h_{ab})-h(W_a,\nabla _TW_b)-h([T,W_a],W_b)\\&=T(h_{ab})-h(W_a,\tau _{*}W_b+[T,W_b])-h([T,W_a],W_b)\\&=T(h_{ab})-A_{ba}-h([W_a,[T,W_b])-h([T,W_a],W_b). \end{aligned}$$

Since the tensor A is self-adjoint by Lemma 2.1, we get

$$\begin{aligned} A_{ab}=\frac{1}{2}\bigg (T(h_{ab})-h([W_a,[T,W_b])-h([T,W_a],W_b)\bigg ). \end{aligned}$$

In particular, $A_{\alpha \beta }=-\frac{1}{2}\big (h(W_{\alpha },[T,W_{\beta }]))+h([T,W_{\alpha }],W_{\beta })\big )$. Applying Lemma A.2 with respect to the frame $\{W_a,\widehat{T}\}$, we get

$$\begin{aligned} \widehat{A}_{\alpha \beta }&=-\frac{1}{2}\bigg (\widehat{h}(W_{\alpha },[\widehat{T},W_{\beta }]))+\widehat{h}([\widehat{T},W_{\alpha }],W_{\beta })\bigg ) {=}A_{\alpha \beta }-\frac{i}{2}Z_{\alpha }Z_{\beta }u-\frac{i}{2}Z_{\beta }Z_{\alpha }u{+}\mathscr {O}_{m-1}\\&=A_{\alpha \beta }-iZ_{\alpha }Z_{\beta }u+\mathscr {O}_{m-1}, \end{aligned}$$

by $\widehat{h}=(1+ \mathscr {O}_m)h$, (A.5) and $[Z_{\alpha },Z_{\beta }]=0$. By (2.11) with respect to frame $\{W_a,\widehat{T}\}$, we get

$$\begin{aligned} \widehat{R}_{\alpha \ \gamma \bar{\beta }}^{\ \gamma }=W_{\gamma }\widehat{\Gamma }_{\bar{\beta }\alpha }^{\ \gamma }-W_{\bar{\beta }}\widehat{\Gamma }_{\gamma \alpha }^{\gamma } -\widehat{\Gamma }_{\gamma \bar{\beta }}^e\widehat{\Gamma }_{e\alpha }^{\gamma } +\widehat{\Gamma }_{\bar{\beta }\gamma }^e\widehat{\Gamma }_{e\alpha }^{\gamma } -\widehat{\Gamma }_{\gamma \alpha }^e\widehat{\Gamma }_{\bar{\beta }e}^{\gamma } +\widehat{\Gamma }_{\bar{\beta }\alpha }^e\widehat{\Gamma }_{{\gamma }e}^{\gamma } +2\widehat{\Gamma }_{\widehat{0}\alpha }^{\gamma }\widehat{J}_{\gamma \bar{\beta }}. \end{aligned}$$

(A.8)

By the first identity in (A.6) and (2.14), for $u\in \mathscr {O}_m$, we have

$$\begin{aligned} W_{\gamma }\widehat{\Gamma }_{\bar{\beta }\alpha }^{\ \gamma }&=W_\gamma \bigg (\Gamma _{\bar{\beta }\alpha }^{\ \gamma }+u_{\bar{\beta }}\delta _{\alpha }^{\gamma }-u^{\gamma }h_{\alpha \bar{\beta }}\bigg )=W_\gamma \Gamma _{\bar{\beta }\alpha }^{\ \gamma }+W_{\gamma }(u_{\bar{\beta }})\delta _{\alpha }^{\gamma }-h^{\gamma \bar{\mu }}W_\gamma (u_{\bar{\mu }})h_{\alpha \bar{\beta }}\nonumber \\&=W_\gamma \Gamma _{\bar{\beta }\alpha }^{\ \gamma } +Z_{\alpha }Z_{\bar{\beta }}u-\delta _{\alpha \bar{\beta }}Z_{\gamma }Z_{\bar{\gamma }}u+\mathscr {O}_{m-1}, \end{aligned}$$

(A.9)

and

$$\begin{aligned} W_{\bar{\beta }}\widehat{\Gamma }_{\gamma \alpha }^{\gamma }&= W_{\bar{\beta }}\bigg ({\Gamma }_{\gamma \alpha }^{\gamma } +u_{\gamma }\delta _{\alpha }^{\gamma }+u_{\alpha }\delta _{\gamma }^{\gamma }\bigg ) =W_{\bar{\beta }}{\Gamma }_{\gamma \alpha }^{\gamma }+(n+1) Z_{\bar{\beta }}Z_{\alpha }u+\mathscr {O}_{m-1}. \end{aligned}$$

Again by the first identity of (A.6), we have $\widehat{\Gamma }_{ab}^c={\Gamma }_{ab}^c+\mathscr {O}_{m-1}$, and by (2.27), we have ${\Gamma }_{ab}^c=\mathscr {O}_{1}$. So we get

$$\begin{aligned} \widehat{\Gamma }_{ab}^c\widehat{\Gamma }_{de}^f={\Gamma }_{ab}^c{\Gamma }_{de}^f+\mathscr {O}_{m}, \end{aligned}$$

for any indices a, b, c, d, e, f. By the second identity of (A.6), $J_{\gamma \bar{\beta }}=-i\delta _{\gamma \bar{\beta }}$ in (2.14) and $u_0=Tu=\frac{\partial {u}}{\partial {t}}+\mathscr {O}_m$ by (2.26), we get

$$\begin{aligned} 2\widehat{\Gamma }_{\widehat{0}\alpha }^{\gamma }\widehat{J}_{\gamma \bar{\beta }}&=2\bigg (\Gamma _{0\alpha }^{\gamma }+u_0\delta _{\alpha }^{\gamma } -\frac{i}{2}Z_{\bar{\gamma }}Z_{\alpha }u-\frac{i}{2}Z_{\alpha }Z_{\bar{\gamma }}u\bigg )J_{\gamma \bar{\beta }}+\mathscr {O}_{m-1}\nonumber \\&=2\Gamma _{0\alpha }^{\gamma }J_{\gamma \bar{\beta }}-2i\frac{\partial {u}}{\partial {t}}\delta _{\alpha \bar{\beta }} -Z_{\bar{\beta }}Z_{\alpha }u-Z_{\alpha }Z_{\bar{\beta }}u+\mathscr {O}_{m-1}. \end{aligned}$$

(A.10)

Noting that $[Z_{\alpha },Z_{\bar{\beta }}] =2i\delta _{\alpha \bar{\beta }}\frac{\partial }{\partial {t}}$, (A.8) leads to

$$\begin{aligned} \widehat{R}_{\alpha \ \gamma \bar{\beta }}^{\ \gamma }&=R_{\alpha \ \gamma \bar{\beta }}^{\ \gamma }-2i\delta _{\alpha \bar{\beta }}\frac{\partial {u}}{\partial {t}}-(n+2) Z_{\bar{\beta }}Z_{\alpha }u -\delta _{\alpha \bar{\beta }}Z_{\gamma }Z_{\bar{\gamma }}u+\mathscr {O}_{m-1}\\&=R_{\alpha \ \gamma \bar{\beta }}^{\ \gamma }-\frac{n+2}{2}\bigg (Z_{\bar{\beta }} Z_{\alpha }u+Z_{\alpha }Z_{\bar{\beta }}u\bigg ) +\frac{1}{2}{\delta }_{\alpha \bar{\beta }}\mathscr {L}_0u+\mathscr {O}_{m-1}, \end{aligned}$$

with $\mathscr {L}_0=-(Z_{\alpha }Z_{\bar{\alpha }}+Z_{\bar{\alpha }}Z_{\alpha })$. $\square$

Appendix B: Calculation of $a_2(n)$ and $b_2(n)$

Recall that we choose special frames satisfying Proposition 5.1 over a contact Riemannian manifold $(M,\theta ,h,J)$.

1.1 B.1. Calculation of $v_2^{jk}$

Lemma B.1

For $v_2^{jk}$ defined in (5.4), we have

$$\begin{aligned} \begin{aligned} v_2^{\alpha \gamma }&=-\frac{1}{6}R_{d\ c\bar{\alpha }}^{\ \gamma }(q)z^dz^c,\qquad \qquad \qquad \qquad \qquad \qquad v_2^{\bar{\alpha }\bar{\gamma }}=-\frac{1}{6}R_{d\ c\gamma }^{\ \bar{\alpha }}(q)z^dz^c,\\ v_2^{\alpha \bar{\gamma }}&=-\frac{1}{6}\bigg (R_{d\ c\gamma }^{\ \alpha }(q)+R_{d\ c\bar{\alpha }}^{\ \bar{\gamma }}(q)\bigg )z^cz^d+\delta ^{\alpha }_{\gamma }v_2,\qquad \quad v_2^{\bar{\alpha }\gamma }=0,\\ v_2^{\alpha 0}&=-\frac{1}{2}J_{b\bar{\alpha }(2)}z^{b} +\frac{i}{12}R_{d\ c\bar{\alpha }}^{\ \bar{\rho }}(q)z^dz^cz^{\rho }-\frac{i}{12}R_{d\ c\bar{\alpha }}^{\ \rho }(q)z^dz^cz^{\bar{\rho }},\\ v_2^{0\bar{\alpha }}&=-\frac{1}{2}J_{b\alpha (2)}z^{b} +\frac{i}{12}R_{d\ c\alpha }^{\ \bar{\rho }}(q)z^dz^cz^{\rho }-\frac{i}{12}R_{d\ c\alpha }^{\ \rho }(q)z^dz^cz^{\bar{\rho }},\\ v_2^{\bar{\alpha }0}&=0=v_2^{0\alpha }, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad v_2^{00}=\frac{4}{9}Q_{\gamma \lambda }^{\bar{\beta }}(q)Q_{\bar{\sigma }\bar{\mu }}^{\beta }(q)z^{\gamma }z^{\lambda }z^{\bar{\sigma }}z^{\bar{\mu }}. \end{aligned} \end{aligned}$$

(B.1)

Proof

In the sequel, we will use Proposition 2.8 repeatedly, especially $s_{\beta (0)}^{\alpha }=s_{\bar{\beta }(0)}^{\bar{\alpha }}=\delta _{\beta }^{\alpha }, s_{\beta (0)}^{\bar{\alpha }}=s_{\bar{\beta }(0)}^{\alpha }=0, s_{b(1)}^j=0, s_{b(0)}^0 =0,$ and we also have $v_0=1$, $v_1=0$, by Corollary 5.1. We find that

$$\begin{aligned} v_2^{\alpha \gamma }&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1)}^{\alpha }s_{\bar{\beta }(m_2)}^{\gamma }v_{m_0}\\&=s_{\beta (2)}^{\alpha }s_{\bar{\beta }(0)}^{\gamma }v_{0}+s_{\beta (0)}^{\alpha }s_{\bar{\beta }(2)}^{\gamma }v_{0} +s_{\beta (1)}^{\alpha }s_{\bar{\beta }(1)}^{\gamma }v_{0}+s_{\beta (1)}^{\alpha }s_{\bar{\beta }(0)}^{\gamma }v_{1} +s_{\beta (0)}^{\alpha }s_{\bar{\beta }(1)}^{\gamma }v_{1}+s_{\beta (0)}^{\alpha }s_{\bar{\beta }(0)}^{\gamma }v_{2}\\&=\delta _{\beta }^{\alpha }s_{\bar{\beta }(2)}^{\gamma }v_0=s_{\bar{\alpha }(2)}^{\gamma } =-\frac{1}{6}R_{d\ c\bar{\alpha }}^{\ \gamma }(q)z^dz^c, \end{aligned}$$

by (2.24) for $s_{\bar{\beta }(2)}^{\alpha }$. Similarly we get

$$\begin{aligned} v_2^{\alpha \bar{\gamma }}&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1)}^{\alpha }s_{\bar{\beta }(m_2)}^{\bar{\gamma }}v_{m_0} =s_{\beta (2)}^{\alpha }\delta _{\bar{\beta }}^{\bar{\gamma }}+\delta _{\beta }^{\alpha }s_{\bar{\beta }(2)}^{\bar{\gamma }} +\delta _{\beta }^{\alpha }\delta _{\bar{\beta }}^{\bar{\gamma }}v_2 =s_{\gamma (2)}^{\alpha }+s_{\bar{\alpha }(2)}^{\bar{\gamma }}+\delta _{\gamma }^{\alpha }v_2\\&=-\frac{1}{6}\bigg (R_{d\ c\gamma }^{\ \alpha }(q)+R_{d\ c\bar{\alpha }}^{\ \bar{\gamma }}(q)\bigg )z^dz^c+\delta _{\gamma }^{\alpha }v_2,\\ v_2^{\bar{\alpha }\gamma }&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1)}^{\bar{\alpha }}s_{\bar{\beta }(m_2)}^{\gamma }v_{m_0}=0,\\ v_2^{\bar{\alpha }\bar{\gamma }}&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1)}^{\bar{\alpha }}s_{\bar{\beta }(m_2)}^{\bar{\gamma }}v_{m_0} =s_{\beta (2)}^{\bar{\alpha }}\delta _{\bar{\beta }}^{\bar{\gamma }}=s_{\gamma (2)}^{\bar{\alpha }} =-\frac{1}{6}R_{d\ c\gamma }^{\ \bar{\alpha }}(q)z^dz^c,\\ v_2^{\alpha 0}&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1)}^{\alpha }s_{\bar{\beta }(m_2+1)}^0v_{m_0} =\delta _{\beta }^{\alpha }s_{\bar{\beta }(3)}^0=s_{\bar{\alpha }(3)}^0\\&=-\frac{1}{2}J_{b\bar{\alpha }(2)}z^{b} +\frac{i}{12}R_{d\ c\bar{\alpha }}^{\ \bar{\rho }}(q)z^dz^cz^{\rho }-\frac{i}{12}R_{d\ c\bar{\alpha }}^{\ \rho }(q)z^dz^cz^{\bar{\rho }},\\ v_2^{\bar{\alpha }0}&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1)}^{\bar{\alpha }}s_{\bar{\beta }(m_2+1)}^0v_{m_0}=0,\\ v_2^{0\alpha }&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1+1)}^0s_{\bar{\beta }(m_2)}^{\alpha }v_{m_0}=0,\\ v_2^{0\bar{\alpha }}&=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1+1)}^0s_{\bar{\beta }(m_2)}^{\bar{\alpha }}v_{m_0} =s_{\beta (3)}^0\delta _{\bar{\beta }}^{\bar{\alpha }}=s_{\alpha (3)}^0\\&=-\frac{1}{2}J_{b\alpha (2)}z^{b} +\frac{i}{12}R_{d\ c\alpha }^{\ \bar{\rho }}(q)z^dz^cz^{\rho }-\frac{i}{12}R_{d\ c\alpha }^{\ \rho }(q)z^dz^cz^{\bar{\rho }}. \end{aligned}$$

By Corollary 3.1 for $s_{b(2)}^0$ and Proposition 3.3, we get $v_2^{00}=\sum \limits _{m_0+m_1+m_2=2,\beta }s_{\beta (m_1+1)}^0s_{\bar{\beta }(m_2+1)}^0v_{m_0} =s_{\beta (2)}^0s_{\bar{\beta }(2)}^0 =\frac{4}{9}Q_{\gamma \lambda }^{\bar{\beta }}(q)Q_{\bar{\sigma }\bar{\mu }}^{\beta }(q)z^{\gamma }z^{\lambda }z^{\bar{\sigma }}z^{\bar{\mu }}.$ So we finish the proof of Lemma B.1. $\square$

1.2 B.2 Proof of Lemma 5.7

By the first identity in (B.1) for $v_2^{\alpha \gamma }$ and (5.5), we get

$$\begin{aligned} \int _{\mathscr {H}^n}v_2^{\alpha \gamma }Z_{\alpha }\Phi {Z_{\gamma }\Phi }dV&=\int _{\mathscr {H}^n}\frac{n^2}{6}R_{d\ c\bar{\alpha }}^{\ \gamma }(q)z^dz^{\bar{\gamma }}z^cz^{\bar{\alpha }} \frac{t^2+2i(|z|^2+1)t-(|z|^2+1)^2}{|w+i|^{2n+4}}dV\nonumber \\&=\int _{\mathscr {H}^n}\frac{n^2}{6}R_{\rho \ \lambda \bar{\alpha }}^{\ \gamma }(q) z^{\rho }z^{\bar{\gamma }}z^{\lambda }z^{\bar{\alpha }} \frac{t^2+2i(|z|^2+1)t-(|z|^2+1)^2}{|w+i|^{2n+4}}dV\nonumber \\&=\frac{n^2(4\pi )^n}{12(n+1)}\mathfrak {Q} \int _{-\infty }^{\infty }\int _0^{\infty }\frac{t^2+2i(r^2+1) t-(r^2+1)^2}{|1+i(1+r^2)|^{2n+4}}r^{2n+3}drdt, \end{aligned}$$

(B.2)

where the last identity follows from the third identity in (5.8). Similarly we have

$$\begin{aligned} \int _{\mathscr {H}^n}v_2^{\bar{\alpha }\bar{\gamma }}Z_{\bar{\alpha }}\Phi {Z_{\bar{\gamma }}\Phi }dV =\frac{n^2(4\pi )^n}{12(n+1)}\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty }\frac{t^2-2i(r^2+1)t -(r^2+1)^2}{|1+i(1+r^2)|^{2n+4}}r^{2n+3}drdt. \end{aligned}$$

(B.3)

Then by (B.1) and (5.5), we get

$$\begin{aligned} \begin{aligned}&\int _{\mathscr {H}^n}v_2^{\alpha \bar{\gamma }}Z_{\alpha }\Phi {Z_{\bar{\gamma }}\Phi }dV\\&\quad =\int _{\mathscr {H}^n}\left[ -\frac{n^2}{6}\left( R_{d\ c\gamma }^{\ \alpha }(q) +R_{d\ c\bar{\alpha }}^{\ \bar{\gamma }}(q)\right) z^dz^c +\delta _{\gamma }^{\alpha }v_2\right] z^{\bar{\alpha }}z^{\gamma }\frac{t^2+(|z|^2+1)^2}{|w+i|^{2n+4}}dV\\&\quad =\int _{\mathscr {H}^n}\bigg [\frac{n^2}{6}\left( R_{\rho \ \gamma \bar{\lambda }}^{\ \alpha }(q)z^{\rho }z^{\bar{\alpha }}z^{\gamma }z^{\bar{\lambda }}-R_{\bar{\rho }\ \lambda \gamma }^{\ \alpha }(q)z^{\bar{\rho }}z^{\bar{\alpha }}z^{\lambda }z^{\gamma } -R_{\rho \ \bar{\lambda }\bar{\alpha }}^{\ \bar{\gamma }}(q)z^{\rho }z^{\gamma }z^{\bar{\lambda }}z^{\bar{\alpha }}\right. \\&\qquad \left. +R_{\bar{\rho }\ \bar{\alpha }\lambda }^{\ \bar{\gamma }}(q)z^{\bar{\rho }}z^{\gamma }z^{\bar{\alpha }}z^{\lambda }\right) \frac{t^2+(|z|^2+1)^2}{|w+i|^{2n+4}}+v_2n^2|z|^2\frac{t^2+(|z|^2+1)^2}{|w+i|^{2n+4}}\bigg ]dV\\&\quad =\frac{n^2}{6(n+1)}(4\pi )^n\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty } \frac{\big (t^2+(r^2+1)^2\big )r^{2n+3}}{|t^2+i(1+r^2)|^{2n+4}}drdt\\&\qquad +\frac{n^2}{6}(4\pi )^n\mathfrak {Q}N_1(2n+2,2n+3,0). \end{aligned} \end{aligned}$$

(B.4)

The last identity follows from the third and fourth identities in (5.8) and

$$\begin{aligned}&\int _{\mathscr {H}^n}v_2n^2|z|^2\frac{t^2+(|z|^2+1)^2}{|w+i|^{2n+4}}dV\\&\quad =-\frac{1}{6}\int _{\mathscr {H}^n}\left( R_{\bar{\beta }\ \alpha \mu }^{\ \alpha }(q)z^{\bar{\beta }}z^{\mu }+R_{\beta \ \bar{\alpha }\bar{\mu }}^{\ \bar{\alpha }}(q)z^{\beta }z^{\bar{\mu }}\right) n^2|z|^2\frac{t^2+(|z|^2+1)^2}{|w+i|^{2n+4}}dV\\&\quad =\frac{n^2}{6}(4\pi )^n\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty }\frac{r^{2n+3}}{|t+i(1+r^2)|^{2n+2}}drdt=\frac{n^2}{6}(4\pi )^n\mathfrak {Q}N_1(2n+2,2n+3,0), \end{aligned}$$

by (5.6) for $v_2$, (5.8) and Lemma 5.4. And by (B.1), we have

$$\begin{aligned} \int _{\mathscr {H}^n}v_2^{\bar{\alpha }\gamma }Z_{\bar{\alpha }}{\Phi }Z_{\gamma }{\Phi }dV=0. \end{aligned}$$

(B.5)

Taking summation of (B.2)–(B.5), we get

$$\begin{aligned}&\int _{\mathscr {H}^n}v_2^{ab}Z_a{\Phi }Z_b{\Phi }dV\\&\quad =\frac{n^2}{3(n+1)}(4\pi )^n\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty }\frac{t^2r^{2n+3}}{|t^2+i(1+r^2)|^{2n+4}}drdt\\&\quad +\frac{n^2}{6}(4\pi )^n\mathfrak {Q}N_1(2n+2,2n+3,0)\\&\quad =\frac{n^2}{3(n+1)}(4\pi )^nN_1(2n+4,2n+3,2)\mathfrak {Q}+\frac{n^2}{6}(4\pi )^nN_1(2n+2,2n+3,0)\mathfrak {Q}\\&\quad =\frac{n^4+2n^3+2n^2}{6(n-1)(n+1)}(4\pi )^nN_1(2n+2,2n+1,0)\mathfrak {Q}, \end{aligned}$$

by using the last two identities in (5.9). So the first identity in (5.17) follows.

By (B.1), we get

$$\begin{aligned} \int _{\mathscr {H}^n}v_2^{0\alpha }Z_0{\Phi }Z_{\alpha }{\Phi }dV=\int _{\mathscr {H}^n}v_2^{\bar{\alpha }0}Z_{\bar{\alpha }}{\Phi }Z_0{\Phi }dV=0. \end{aligned}$$

(B.6)

By (B.1), (5.5) and substituting identities in (5.8) for certain terms, we get

$$\begin{aligned}&\int _{\mathscr {H}^n} v_2^{\alpha 0}Z_{\alpha }{\Phi }Z_0{\Phi }dV =\int _{\mathscr {H}^n}\bigg (-\frac{1}{2}J_{b\bar{\alpha }(2)}z^{b} +\frac{i}{12}R_{d\ c\bar{\alpha }}^{\ \bar{\rho }}(q)z^cz^dz^{\rho }-\frac{i}{12}R_{d\ c\bar{\alpha }}^{\ \rho }(q)z^cz^dz^{\bar{\rho }}\bigg )\nonumber \\&\qquad \cdot (-in^2z^{\bar{\alpha }})\frac{t^2+it(|z|^2+1)}{|w+i|^{2n+4}}dV\nonumber \\&\quad =n^2\int _{\mathscr {H}^n}\bigg (\frac{i}{2}J_{\beta \bar{\alpha }(2)}z^{\beta }z^{\bar{\alpha }}+\frac{i}{2}J_{\bar{\beta }\bar{\alpha }(2)}z^{\bar{\beta }}z^{\bar{\alpha }} +\frac{1}{12}\big (R_{\beta \ \bar{\mu }\bar{\alpha }}^{\ \bar{\rho }}(q)z^{\beta }z^{\rho }z^{\bar{\mu }}z^{\bar{\alpha }}-R_{\bar{\mu }\ \bar{\alpha }\beta }^{\ \bar{\rho }}(q)z^{\bar{\mu }}z^{\rho }z^{\bar{\alpha }}z^{\beta }\big )\nonumber \\&\qquad -\frac{1}{12}R_{\beta \ \mu \bar{\alpha }}^{\ \rho }(q)z^{\beta }z^{\bar{\rho }}z^{\mu }z^{\bar{\alpha }}\bigg )\frac{t^2+it(|z|^2+1)}{|w+i|^{2n+4}}dV\nonumber \\&\quad =\frac{n^2}{n+1}(4\pi )^n\bigg (-\frac{3}{4}+0+0-\frac{1}{24}-\frac{1}{24}\bigg )\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty }\frac{t^2+it(r^2+1)}{|t^2+i(1+r^2)|^{2n+4}}r^{2n+3}drdt\nonumber \\&\quad =-\frac{5n^2}{6(n+1)}(4\pi )^n\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty }\frac{t^2+it(r^2+1)}{|t^2+i(1+r^2)|^{2n+4}}r^{2n+3}drdt. \end{aligned}$$

(B.7)

By taking conjugation of (B.7), we get

$$\begin{aligned} \int _{\mathscr {H}^n}v_2^{0\bar{\alpha }}Z_0{\Phi }Z_{\bar{\alpha }}{\Phi }dV =-\frac{5n^2}{6(n+1)}(4\pi )^n\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty }\frac{t^2-it(r^2+1)}{|t^2+i(1+r^2)|^{2n+4}}r^{2n+3}drdt. \end{aligned}$$

(B.8)

Now taking summation of (B.6)–(B.8), we get

$$\begin{aligned}&\int _{\mathscr {H}^n}\left( v_2^{a0}Z_a{\Phi }Z_0{\Phi }+v_2^{0a}Z_0{\Phi }Z_a{\Phi }\right) dV\\&\quad =-\frac{5n^2}{3(n+1)}(4\pi )^n\mathfrak {Q}\int _{-\infty }^{\infty }\int _0^{\infty }\frac{r^{2n+3}t^2}{|t^2+i(1+r^2)|^{2n+4}}drdt\\&\quad =-\frac{5n^2}{3(n+1)}(4\pi )^n\mathfrak {Q}N_1(2n+4,2n+3,2)=-\frac{5n^2}{6(n+1)(n-1)}\mathfrak {Q}N_1(2n+2,2n+1,0), \end{aligned}$$

by (5.9) and Lemma 5.4. So the second identity in (5.17) follows.

By (B.1), (5.9), (5.5), the fifth identity in (5.8) and Lemma 5.4, we get

$$\begin{aligned}&\int _{\mathscr {H}^n}v_2^{00}Z_0{\Phi }Z_0{\Phi }dV=\int _{\mathscr {H}^n} \frac{4}{9}Q_{\gamma \lambda }^{\bar{\beta }}(q)Q_{\bar{\sigma } \bar{\mu }}^{\beta }(q)z^{\gamma }z^{\lambda }z^{\bar{\sigma }}z^{\bar{\mu }} \frac{n^2t^2}{|w+i|^{2n+4}}dV\\&\quad =\int _{-\infty }^{\infty }\int _0^{\infty }\frac{4n^2}{3(n+1)}(4\pi )^n \mathfrak {Q}\frac{r^{2n+3}t^2}{|t^2+i(1+r^2)|^{2n+4}}drdt\\&\quad =\frac{4n^2}{3(n+1)}(4\pi )^nN_1(2n+4,2n+3,2)\mathfrak {Q} =\frac{2n^2}{3(n+1)(n-1)}(4\pi )^nN_1(2n+2,2n+1,0)\mathfrak {Q}. \end{aligned}$$

So the third identity in (5.17) follows.