I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.

I am having some problems with both the notation and the geometrical side.

1)I don't know what kind of objects $N,L$ are precisely. In Lemma 1.1 the author says that $L$ is a smooth $(1,0)$-tangent vector field on $\partial \Omega$, so I think it should be something like $$ L=\sum_{j=1}^na_j\frac{\partial}{\partial z_j} $$ where $a_j$'a are some smooth functions. Are they complex valued?

2) At the end of page 3, he writes $$ \operatorname{Hess}_{\rho}(L,N)=g(\nabla_L\nabla r,N) $$ where $g$ stands for the euclidean metric in $\Bbb C^n$. Thus $g$ should represent a standard scalar vector between its first entry and the conjugate of the second, that is $$ \nabla_L\nabla r\cdot \overline N $$ am I right? What does $$ \nabla_L $$ mean?

3)Let us look at the proof of Lemma 1.1; for what reason $N_r-\overline{N_r}$ should be tangent to $\partial\Omega$? How should I check this? Why does this imply $(N_r-\overline{N_r})\overline L\rho=0$? Does "$z$ approaches $p$ from the normal direction" play any relevant role in the computations?

I would have other questions about it, but I hope these questions could help you to understand my gaps, and hopefully give an answer or some suitable reference.

I know my questions are basics, but the paper is MO's stuff, I think, that's why I wrote here instead MSE.