Complex affine spheres and a Bers theorem for SL(3,C)

Abstract: 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 $\mathrm{T}(S)\times \overline{\mathrm{T}(S)}$. The open subset contains $\mathrm{Hit}_3(S)\times \overline{\mathrm{T}(S)}$ and $\mathrm{T}(S)\times \overline{\mathrm{Hit}_3(S)}$, 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.