We denote

(4.129)

$$\hat{h}_{-}=A_{-}-\int_{T_{-}}^{0}h(u)du-\frac{1}{2}\log\|S_{H}\|.$$

By the Poincaré-Lelong equation we have

(4.130)

$$d_{D}d_{D}^{c}\log{\|S_{H}\|}^{2}=4\pi\delta_{H}-(k_{-}-k_{+})\omega_{D},$$

where $\delta_{H}$ denotes the current of integration along $H$. By directly taking derivatives and use (2.13) we obtain that outside $H$,

(4.131)

$$d_{D}d_{D}^{c}(\int_{0}^{T_{-}}h(z)dz)=\int_{0}^{T_{-}}d_{D}d_{D}^{c}h(z)dz=\int_{0}^{T_{-}}-\partial_{z}^{2}\tilde{\omega}(z)dz=-\partial_{z}\tilde{\omega}|_{z=T_{-}}+\partial_{z}\tilde{\omega}|_{z=0}$$

By (3.349) and (3.381), the right hand side is given by $-\frac{1}{2}(k_{-}-k_{+})\omega_{D}+\epsilon_{T}$. Now using the asymptotics of $h$ near $P$ in (4.17), one sees that $\hat{h}_{-}$ is bounded near $H$. So the following current equation holds globally on $D$

(4.132)

$$d_{D}d_{D}^{c}\hat{h}_{-}=\epsilon_{T}$$

Now

(4.133)

$$\int_{D}\hat{h}_{-}\omega_{D}^{n-1}=A_{-}\int_{D}\omega_{D}^{n-1}+\int_{D}\int_{0}^{T_{-}}h\omega_{D}^{n-1}dz=B_{T}$$

So by standard elliptic regularity we get the conclusion for $\hat{h}_{-}$. The proof for the other equation is similar. ∎