If $P\subset Q$ is an embedded submanifold of co-dimension $3$, then for any $p\in P$, there is a neighborhood $\mathcal{U}$ of $p$ in $Q$, and a Green’s current $G_{U}$ for $U=\mathcal{U}\cap P$ in $\mathcal{U}$ satisfying

(3.46)

$\displaystyle\Delta G_{U}=2\pi\cdot\delta_{U}\ \ \ \ \text{in}\ \ \ \ \ \mathcal{U},$

and the local expansion

	$\displaystyle G_{U}=$	$\displaystyle\frac{1}{2r}(1-\frac{H^{\alpha}y_{\alpha}}{2})dy_{1}\wedge dy_{2}\wedge dy_{3}+\frac{1}{2r}y_{\beta}A_{i\alpha\beta}dx_{i}\wedge dy_{\widehat{\alpha}}-\frac{1}{4}A_{ij\alpha\beta}r\cdot dy_{\widehat{\alpha\beta}}\wedge dx_{i}\wedge dx_{j}$
	$\displaystyle+$	$\displaystyle\frac{3}{16}(A_{i\alpha,\alpha+1}A_{j\alpha,\alpha+2}-A_{i\alpha,\alpha+2}A_{j\alpha,\alpha+1})d(ry_{\alpha})\wedge dx_{i}\wedge dx_{j}+r^{-3}\Pi_{3}^{(4)}+O^{\prime}(r^{2}),$

where $\overrightarrow{H}=H^{\alpha}e_{\alpha}$ is the mean curvature of $P\subset Q$ and the $3$-form $\Pi_{3}^{(4)}$ is defined in (3.45).