g\right)(\n^D\phi, \n^D \phi)-2V^{-\g}\<\n^D \phi, \n^D(V^\g \phi_\nu)\>\right]dA.\nonumber
    \end{aligned}$$

Using (\[div\]), we have $$\begin{aligned}
        &\di\int_{\O}V^\tau D_i\Big((\nb^D \phi)^i D_j((\nb^D \phi)^j)-(\nb^D \phi)^j D_j((\nb^D \phi)^i)\Big) d\O\nonumber\\
        =&\di\int_{\O}\bar\nabla_i\Big[V^\tau\Big((\nb^D \phi)^i D_j((\nb^D \phi)^j)-(\nb^D \phi)^j D_j((\nb^D \phi)^i)\Big)\Big] d\O\nonumber\\
        =&\di\int_{\S}V^{\tau} \Big(\<\bn^D\phi,\nu\>D_j((\nb^D \phi)^j)-V^{\g-\alpha}\phi^j\<D_j(\nb^D \phi),\nu\>\Big) dA.
    \end{aligned}$$ Thus we only need to simplify the boundary term. At any fixed point $P\in \S$, we choose normal coordinates with respect to $\bar g$ such that the indices $a=1,\cdots, n-1$ represents coordinates on $\S$ and $j=1,\cdots,n-1, \nu$ for coordinates on $\O$. For simplicity, we will not distinguish upper and lower indexes and we denote $\phi_{ij}$ as the Hessian $\phi$ with respect to the Levi-Civita connection.

Using (\[Bochner equ 2\]), we obtain $$\begin{aligned}
       & \<\bn^D\phi,\nu\>D_j((\nb^D \phi)^j)
        =V^{2(\g-\a)}\big(\bar\De\phi\phi_\nu+(2\g+n\a)\<\bn u, \bn \phi\>\phi_\nu\big),\label{theo RF equ 2}
    \end{aligned}$$ $$\begin{aligned}
        &V^{\g-\a}\phi^j\<D_j((\nb^D \phi)),\nu\>=V^{2(\g-\a)}\big(\phi_{j\nu}\phi_j+\g u_\nu\phi_j^2+(\a+\g) \<\bn u, \bn \phi\> \phi_\nu\big).
        \label{theo RF equ 3}
    \end{aligned}$$ Combining (\[theo RF equ 2\]) and (\[theo RF equ 3\]), we have $$\begin{aligned}
 \label{theo RF equ 5}
    &\<\bn^D\phi,\nu\>D_j((\nb^D \phi)^j)-V^{\g-\a}\phi^j\<D_j(\nb^D \phi),\nu\>\\
       =&V^{2(\g-\a)}\Big(\phi_\nu\phi_{aa}-\phi_a\phi_{a\nu}+(n-1)\a u_\nu\phi_\nu^2+(\g+(n-1)\a) u_a\phi_a\phi_\nu-\g\phi_a^2u_\nu\Big)\nonumber\\
       =&V^{2(\g-\a)}\Big(H\phi_\nu^2+\phi_ah_{ab}\phi_b+\phi_\nu\De\phi-\phi_a\n_a\phi_\nu\nonumber\\
       &+(n-1)\a u_\nu\phi_\nu^2+(\g+(n-1)\a) u_a\phi_a\phi_\nu-\g\phi_a^2u_\nu\Big),\nonumber  
    \end{aligned}$$ where we used in the last equality the Gauss-Weigarten formula $$\begin{aligned}
    \phi_{aa}=\De\phi+H\phi_\nu,\qquad \phi_{a\nu}=\n_a\phi_\nu-h_{ab}\phi_b.\end{aligned}$$ Using $H^D=H+(n-1)\a u_\nu$ in , we get $$\begin{aligned}
 \label{theo RF equ 6}
        &\di\int_{\S}V^{\tau} \Big(\<\bn^D\phi,\nu\>D_j((\nb^D \phi)^j)-\phi^j\<D_j(\nb^D \phi),\nu\>\Big) dA\\
        =&\di\int_{\S}V^{\tau+2(\g-\a)}\Big(H^D\phi_\nu^2+(h-\g u_\nu g)(\n\phi, \n \phi)\Big)\nonumber\\&+\int_\S V^{\tau+2(\g-\a)}\Big(\phi_\nu\De\phi -\<\n\phi,\n\phi_\nu\>+ (\g+(n-1)\a)\< \n u,\n 