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 Tzitzéica equations. To study this equation we establish analytic results that should be of independent interest.

Removed content on Beltrami equation (now contained in arXiv:2410.06175) and on harmonic maps (to be treated in a different paper)