On a Bers theorem for SL(3,C)

For $S$ a closed surface of genus at least $2$, let $\mathrm{Hit}_3(S)$ be
the Hitchin component of representations to $\mathrm{SL}(3,\mathbb{R}),$
equipped with the Labourie-Loftin complex structure. We construct a mapping
class group equivariant holomorphic map from a large open subset of
$\mathrm{Hit}_3(S)\times \overline{\mathrm{Hit}_3(S)}$ to the
$\mathrm{SL}(3,\mathbb{C})$-character variety that restricts to the identity on
the diagonal and to Bers' simultaneous uniformization on $\mathcal{T}(S)\times
\overline{\mathcal{T}(S)}$. The open subset contains $\mathrm{Hit}_3(S)\times
\overline{\mathcal{T}(S)}$ and $\mathcal{T}(S)\times
\overline{\mathrm{Hit}_3(S)}$, and the image includes the holonomies of
$\mathrm{SL}(3,\mathbb{C})$-opers.
The map is realized by associating pairs of Hitchin representations to
immersions into $\mathbb{C}^3$ that we call complex affine spheres, which are
equivalent to certain conformal harmonic maps into
$\mathrm{SL}(3,\mathbb{C})/\mathrm{SO}(3,\mathbb{C})$ and to new objects called
bi-Higgs bundles. Complex affine spheres are obtained by solving a second-order
complex elliptic PDE that resembles both the Beltrami and Tzitz\'eica
equations. To study this equation we establish analytic results that should be
of independent interest.