Upon reading [this answer][1] to [this question][2], the last paragraph mentions the following. "Requiring the [almost complex] structure to be integrable corresponds to a certain PDE for this  map." The map in question, I presume, can be taken to be some $f: M \to Gr(3, \mathbb{C}^\infty)$ for the almost complex $6$-dimensional manifold $M$, which corresponds to $TM$ by pullback of the tautological bundle.

I'm no expert in these topics so I am struggling to unpack this claim. What is this PDE and how do we obtain it? Furthermore, is there a reference to Yau's mentioned program or a survey of attempts/results in that direction?

**EDIT**

I've emailed Professor Yau and to my delight he responded. Here is what I've learned. The map $f$ above determines an almost complex structure $J$ on $TM$. Then, as Moishe mentioned in the comments, we have that the Newlander-Nirenberg theorem gives the integrable condition through the vanishing of the Nijenhuis tensor $N_J$, which takes as inputs two vector fields $X,Y$. In particular, we may write
$$ N_J : \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM) $$
which is the partial differential operator in question. Yau mentions that $N_J$ can be written in terms of the differential of $f$. This part eludes me.

So my updated question is, how can $N_J$ be written in terms of $df$?


  [1]: https://mathoverflow.net/a/1984/143629
  [2]: https://mathoverflow.net/questions/1973/is-there-a-complex-structure-on-the-6-sphere